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CP Violating Bubble Wall and Electroweak Baryogenesis

The electroweak baryogenesis depends on the profile of the CP-violating bubble wall created at the first order phase transition. We attempt to determine it by solving the coupled equations of motion for the moduli and phases of the two Higgs doublets at the transition temperature. A variety of CP-violating bubble walls are classified by boundary conditions. We point out that a sufficiently small explicit CP violation gives nonperturbative effects to yield the baryon asymmetry of the universe.


Introduction
To generate the baryon asymmetry of the universe (BAU) starting from a baryonsymmetric universe, Sakharov's three conditions must be met. If there was no B − L down to the electroweak era, baryogenesis mechanism should work at the electroweak phase transition (EWPT) for the present universe to be realized [1]. Even if some B − L existed, what happened at the EWPT would affect the present BAU. Among the conditions, to have sufficient CP violation necessary for the BAU will require some extension of the Higgs sector in the minimal standard model. Further, the minimal standard model satisfying the present lower bound on the Higgs mass cannot make the EWPT first order and cannot suppress the sphaleron process after that. Two-Higgs-doublet extension of the standard model is the simplest extension, which can yield the Higgs-sector CP violation, and includes the Higgs sector of the minimal supersymmetric standard model (MSSM) as a special case. Here we concentrate on such two-Higgs-doublet models.
At the first-order EWPT, a bubble wall dividing the broken and symmetric phases is created and grows to convert the whole universe into the broken phase. The profile of the bubble wall will be characterized by the classical configuration of the Higgs and gauge fields. It is essential in any scenario of the electroweak baryogenesis to know the spacetimevarying profile of the bubble wall, in which the relative phase of the Higgs scalars violates CP . In literatures, some functional forms of the profile were assumed [2] [3]. They should be determined, however, by the dynamics of the gauge-Higgs system. That is, the bubble wall will be a solution to the classical equations of motion which has the effective potential as the potential for the Higgs sector. The observable CP violations in our world offer the boundary condition in the broken phase for these equations of motion.
In a previous paper [4], assuming some appropriate form of the effective potential, we found a solution such that CP -violating relative phase becomes as large as O(1) around the wall while it completely vanishes in the broken and symmetric phase limits. In the solution, the moduli were fixed to be the kink shape. Such a solution, which we call as a "solution with kink ansatz", is interesting since it does yield sufficient amount of the chiral charge flux through the wall surface, which will be turned into the baryon number in the symmetric phase region by the sphaleron transition. The solution emerges when a mechanism similar to the spontaneous CP violation operates in the intermediate region.
For such a mechanism to work, some of the parameters in the effective potential are restricted similarly to the case of the spontaneous CP violation at zero temperature, although less constrained at finite temperature. The next task will be to determine the moduli themselves of the Higgs fields dynamically as well as the relative phase of them. In this paper, we solve the coupled equations of motion for the moduli and the relative phase of the two Higgs fields by imposing a discrete symmetry between the two Higgs doublets to reduce the dynamical degrees of freedom. With the same parameters of the effective potential as those with the kink ansatz, we present a solution, which we call as a "solution without kink ansatz", having larger CP -violating phase and non-kink type moduli. As expected, this solution has lower energy than that with the kink ansatz. We also give another class of new solutions, which are caused by a similar mechanism of CP violation in the intermediate region as above but are free from the constraints of zero-temperature spontaneous CP violation. The nontrivial phase dependence of cubic terms in the moduli, which is essential to the first-order EWPT, plays a crucial role in this mechanism. Such terms will be induced from bosonic loops, especially those of the scalar partners of quarks and leptons in the MSSM and the extra Higgs scalars.
If CP violation is completely spontaneous, a CP -violating bubble is created with the same probability as its CP -conjugate one. Then the net baryon number would result in zero. We show that a tiny explicit CP violation, which is consistent with the present bound on the neutron electric dipole moment (EDM), can nonperturbatively resolve the degeneracy between the CP -conjugate pair of the bubbles to leave a sufficient BAU after the EWPT.
In section 2, we postulate the effective potential of the Higgs sector and present the equations of motion. In section 3, we classify possible solutions according to the boundary conditions, and give some examples of "solutions without kink ansatz". We also investigate solutions in the presence of explicit CP breaking, and point out the importance of the breaking to generate the BAU. Section 4 is devoted to conclusion and discussions.

The Model
One of the most attractive features of electroweak baryogenesis is the fact that it depends only on physics which can be tested by experiments at present as well as in the near future. We emphasize that it is CP violation at the first-order EWPT, not that observed in laboratories, which affects the generated baryon asymmetry. The latter is restricted by various observables such as the neutron EDM, while the former is not directly. Here we attempt to relate the observable CP violation to that at the EWPT.

Ansatz for the effective potential
Given a model, one can calculate the effective potential near the phase transition temperature, which determines the order of the EWPT as well as its dynamics. We expect that the bubble wall will be well approximated by a solution to the classical equations of motion derived from the effective Higgs potential, which consists of the tree-level potential with radiative corrections and finite-temperature contributions. Even if the tree-level Higgs potential is CP conserving, the corrected potential parameters may predict spontaneous CP violation, or the corrections may bring effects of various explicit CP -violating parameters, such as those in the soft supersymmetry breaking parameters, into the effective potential. The CP violation in the Kobayashi-Maskawa matrix could also induce CP -violating term in the effective potential through higher order diagrams.
Here we postulate a functional form of the effective potential, which is equipped with the feature of first-order EWPT. We assume that it is well approximated by a gaugeinvariant polynomial up to the fourth order of the moduli of the Higgs fields. The most general gauge-invariant Higgs potential is where m 2 1 , m 2 2 , λ 1 , λ 2 , λ 3 , λ 4 ∈ R and m 2 3 , λ 5 , λ 6 , λ 7 ∈ C, three of their phases are independent and yield the explicit CP violation. To avoid the tree-level Higgs-mediated flavor-changing neutral current (FCNC) interactions naturally, one may impose the discrete symmetry on the lagrangian [5], which is broken by λ 6,7 = 0 and softly by m 2 3 = 0. The Yukawa interactions should have that symmetry not to induce divergent λ 6,7 -terms. In the MSSM, the parameters in the Higgs potential are given by where g (g ′ ) is the SU(2) (U(1)) gauge coupling, the mass parameters are completely arbitrary, and the Yukawa interactions respect the discrete symmetry. Hence CP is not violated by the Higgs sector at the tree level in the MSSM. Nonzero m 2 3 will induce finite corrections to λ 5,6,7 , which violates CP explicitly or spontaneously.
At finite temperature, there arise new kind of corrections, which lead to the first-order EWPT. In general, the one-loop finite-temperature corrections to the effective potential have a formV where A stands for particle species running through the loop, with a A = m A (v)/T , v being the VEV of the Higgs fields, and d A counts the degrees of freedom with sign (e.g., d W = 6, d Z = 3 and d t = −6). One can expand I B,F (a 2 ) around a 2 = 0 to observe that the expansion of I B contains (a 2 ) 3/2 -term, which originates from the Matsubara zero mode of the loop momentum [6]. Hence if m 2 (v) ∼ 0 at v ∼ 0, the effective potential has v 3 terms with the correct sign, which is the characteristic of the first-order phase transition. Here we assume that the effective potential at the EWPT is approximated by the polynomial of the form of (2.1) plus v 3 -terms irrespective of the validity of the perturbation and the high-temperature expansion.
First of all, we shall concentrate on the case with no explicit CP violation in the effective potential. Then the ansatz for it is where we parameterize the Higgs VEV as 1 As we noted, the parameters in (2.5) should be understood to contain radiative and finitetemperature corrections, which can be calculable once the model is specified. According to the high-temperature expansion, B a and C a may arise from the bosonic loops, among which θ-dependent terms, B 1,2 and C 1,2 , could arise only from those of squarks, sleptons and Higgs scalars, that mix scalars and pseudoscalars in the MSSM. Viewed as a function of cos θ, the effective potential is written as (2.7) When Here we assume that U (1) em gauge symmetry is unbroken at the EWPT. and −2m 2 CP is spontaneously broken. At zero temperature, B a = C a = 0 and (ρ 1 , ρ 2 ) = (v 0 cos β 0 , v 0 sin β 0 ) so that these conditions are reduced to the well-known ones, Here all the parameters are evaluated at zero temperature. The former condition (λ 5 < 0) is hardly fulfilled, as long as λ 5 ≥ 0 at the tree level, since almost all radiative corrections except for those by the gauginos and Higgsinos contribute to make λ 5 positive in the MSSM [7]. Even if these conditions are satisfied, such a model will be bothered by a light scalar, which is inevitable in the presence of spontaneous violation of a discrete symmetry [8]. In contrast, the conditions (2.8) and (2.9) can be satisfied in the intermediate range, since ρ 1 (ρ 2 ) varies from v cos β (v sin β) to zero as going from the broken phase to the symmetric phase region. Since the finite-temperature corrections both of bosons and fermions to λ 5 are positive at the one-loop level, requiring (2.8) and (2.9) to be satisfied only near the bubble wall and not to be satisfied in the broken phase would constrain some parameters in the model. Especially in the MSSM, for the finite-temperature corrections to negative λ 5 to be small, some soft-supersymmetry-breaking mass parameters will be bounded from below [9]. If CP is violated spontaneously, there are two kinds of bubbles (ρ i , θ) and (ρ i , −θ) which are CP conjugate with each other and have the same energy density. Such bubbles would be nucleated with the same probability, so that no net baryon number would be generated. An explicit CP violation will resolve this degeneracy and leave the finite BAU. Although various types of explicit CP violation in the effective potential are possible, we restrict to a simple case that only the m 2 3 is complex and the other parameters are real. Hence we replace m 2 3 ρ 1 ρ 2 cos θ in V eff of (2.5) by In the MSSM, nonzero δ is induced from the CP violation in the scalar trilinear terms and |δ| < 10 −2 does not contradict with the present bound on the neutron EDM [10].
The effect of δ on the baryon asymmetry was discussed in [11] perturbatively. We shall see later that δ nonperturbatively discriminates the CP -conjugate pair of the bubbles for some types of solutions.

Equations of motion
We suppose that the dynamics of the bubble wall is governed by the classical equations of motion derived from the lagrangian where, If the bubble is spherically symmetric and sufficiently macroscopic, it is approximated as a planar object so that the system is reduced to one-dimensional. Further, when the bubble grows with a constant velocity keeping the shape of the critical bubble, it can be viewed as a static object. For this effective one-dimensional system, the gauge fields may be written in pure gauge forms and a solution in this case will have lower energy than solutions with nontrivial gauge configuration. Hence we assume that the bubble wall consists of only the Higgs scalar, and work in the gauge where the gauge fields are eliminated. Since the fermions couple to gauge-noninvariant phases of the Higgs instead of the invariant θ, the CP violation in the Yukawa interactions is made unambiguous once the gauge is fixed in this manner. Then the bubble wall profile is characterized by the space-varying order parameters (ρ i (z), θ i (z)), which satisfy the following equations of motion, where z is the coordinate perpendicular to the wall. From the requirement that the gauge fields are pure-gauge type, a constraint equation, "sourcelessness condition", should be fulfilled: Now our task is to solve the equations (2.13) and (2.14) with the constraint (2.15) satisfying certain boundary conditions in the broken phase and symmetric phase regions. Before solving the equations, we reduce the dynamical degrees of freedom by adopting some ansatz. First, we require that the moduli of the Higgs scalars take kink shapes of the common width when θ is sufficiently small. In practice, we assume that when δ = 0, are solutions to (2.13), where 1/a is the common wall thickness. This restricts some of the parameters in the effective potential (2.5) [4]. Then the equation of motion for θ(z) in the background of the kink ρ i (z) is reduced to where we introduced a dimensionless finite-range parameter y by Here θ(z) is related to θ 1,2 (z) by which means that θ 1 (z) and θ 2 (z) satisfy the condition (2.15) when ρ i (z) is the kink shape (2.16). We refer the solution to (2.17) as "the solution with kink ansatz". We found several solutions for various boundary conditions when f = 0 in [4]. Among them, when d < 0 (λ 5 < 0), there is an interesting solution which connects the CP -conserving vacua and largely violates CP in the intermediate range around ρ i (y) ∼ 1/2 by satisfying (2.8) and (2.9). The solution has so large a phase as θ(y = 1/2) > ∼ 0.3 to generate sufficient chiral charge flux. For the same set of parameters, we also found that a tiny explicit CP violation of the form (2.11) can nonperturbatively enhance the lower-energy bubble over its CP -conjugate partner to leave sufficient BAU [12].
The large θ means, however, that the kink ansatz may no longer be adequate. Thus we have to solve coupled equations for the moduli and the phases. In order to simplify the analysis, we impose a discrete symmetry on the effective potential under and assume that it is not spontaneously broken to reduce the dynamical degrees of freedom. This means that the parameters in (2.5) should satisfy and that tan β = 1 by definition. For we have two coupled equations: while the "sourcelessness condition" is automatically satisfied. Here the effective potential is written in terms of the reduced numbers of the parameters as whereλ ≡ λ 1 + λ 3 − λ 4 . The parameters in (2.25) will be further constrained by requiring that V eff should have two degenerate minima corresponding to the broken and symmetric phases. We relate the parameters to those in the potential with the kink ansatz, (b, c, d, e, f ), to compare "the solutions without kink ansatz" to those with it. That is, some of the parameters in (2.25) are related to (b, c, d, e, f ) directly by and from the kink ansatz for the case of θ = 0, the others are given by Once (b, c, d, e, f ), the wall thickness 1/a and the Higgs VEV v at the transition temperature are given, the effective potential is determined. Such an effective potential has the degenerate minima and covers a restricted class of the potentials of the form (2.25). We expect that the solutions obtained below will survive when we slightly deform the effective potential from the restricted class.

Energy density and BAU
The nucleation rate of the bubble is roughly given by exp(−4πR 2 C E/T C ), where R C is the radius of the critical bubble, which is approximated by R C ≃ 3F C /(4πav 2 ). Here F C is the free energy of the critical bubble and is found to be about (140 ∼ 160)T [13]. E is the energy density of the bubble, (2.28) For "the solution with kink ansatz", this is given by where av 2 /3 is the energy density of the trivial kink solution. It was evaluated for several solutions in [4] and [12]. With the discrete symmetry, the energy density is reduced to In the charge transport scenario [2], the generated baryon number n B is proportional to the hypercharge flux F Y , which is caused by the difference of the reflection rates of chiral fermions by the CP -violating bubble wall. That is, where s is the entropy density, N a model-dependent constant of O(1) and κ the coefficient of the sphaleron transition rate in the symmetric phase given by κ = 1.09 ± 0.04 [14]. τ may be approximated by the mean free time or diffusion time D/v w with the diffusion constant D of the charge carrier, where D ≃ 1/T for quarks and D ≃ (10 2 ∼ 10 3 )/T for leptons [15]. It was shown that if the forward scattering is enhanced, even for the top quark, τ T ≃ 10 ∼ 10 3 depending on v w , where the maximum value is realized at v w ≃ 1/ √ 3 [2]. For this optimal case, would be necessary to explain the present BAU. As we noted, if CP is violated spontaneously, the bubble with the profile (ρ i , θ) degenerates with its CP -conjugate partner (ρ i , −θ). Then their nucleation probability and dynamics will be completely the same so that the net baryon number generated is completely canceled with each other. Even if an explicit CP violation will resolve the degeneracy, two (or more) kinds of bubbles contribute to the baryon number in opposite signs. When we denote the baryon number generated by the j-th kind of bubble as (n B /s) j and the nucleation rate of it as N j = exp(−4πR 2 C E j /T C ), the BAU may be approximately given by where the expansion rate of each bubble is assumed to be almost the same. We shall estimate it for some numerical solutions obtained in the next section.

Classification of Bubble Walls and Numerical Examples
We show some solutions to the coupled equations (2.23) and (2.24), that is, "the solutions without kink ansatz". First, one must specify the boundary conditions in the broken and symmetric phases.
Namely, for a fixed v, 2 θ 0 is the solution to As for the symmetric phase, it is obvious that ρ(y = 1) = 0. Although θ 1 ≡ θ(y = 1) seems to be undetermined, it is not the case when m 2 3 = 0 (b = 0). This is because the equation for θ(y), (2.24) written in terms of y, has singularity at y = 1, which makes the solution singular there unless sin(θ 1 + δ) multiplying the most singular m 2 3 -term vanishes. For such a singular solution the energy density diverges so that it cannot be realized dynamically. 3 Hence the boundary condition for θ(y) in the symmetric phase is As long as the effective potential is parameterized as in the previous section and |δ| is sufficiently small (|δ| < 10 −3 ), the possible lowest energy solutions will be classified in terms of the boundary values of θ(y) as in the table below.  Let us denote the solutions of the CP -conjugate pair in the case of δ = 0 as (ρ, θ) and (ρ, −θ). For sufficiently small |δ| = 0, there generally appear three solutions, (ρ + , θ + ), (ρ − , θ − ) and (ρ 0 , θ 0 ), such that they respectively approach (ρ, θ), (ρ, −θ) and the trivial kink solution (ρ kink , θ = 0) as δ → 0. Note that (ρ 0 , θ 0 ) has no CP -conjugate partner. An example presented here employs the same parameter set and the boundary condition as in [12] to solve the equations (2.23) and (2.24) for various δ starting from δ = 0. That is, (b, c, d, e, f ) = (3, 12.2, −2, 12.2, 0) and θ 1 = −δ while θ 0 is determined by (3.2).
There are three numerical solutions, (ρ 0 , θ 0 ), (ρ + , θ + ) and (ρ − , θ − ), in the order of the decreasing energy, satisfying the same boundary conditions for δ < ∼ 0.0045. For larger δ, only (ρ − , θ − ) exists. Their energy densities, measured from that of the trivial kink configuration, δE ≡ E − av 2 /3, are plotted in Fig. 3 Figure 4: Contour plot of the effective potential together with the path of (ρ + , θ + ) connecting the two vacua for the same parameter set as Fig. 3 and δ = 10 −3 .  where we have used the expression for the nucleation rate of the critical bubble with R C ≃ 3F C /(4πav 2 ) and F C ≃ 145T C . By neglecting an extremely small contribution from (ρ 0 , θ 0 ), as understood from the small imaginary part in Fig. 5, it is found that about 88 % of baryon number generated by (ρ − , θ − ) survives the cancellation by (ρ + , θ + ). Thus δ = 10 −3 will be sufficient to leave the BAU for wide range of the carrier mass and the wall thickness. As seen from Fig. 3, only one species of the bubbles exist for δ > 0.0045 so that the cancellation among the bubbles does not occur. The net chiral charge flux (2.33) for δ = 10 −3 is presented in Fig. 6.
Some solutions of type (C) for δ = 0 were found under the the kink ansatz [4]. The profile for non-vanishing δ is similar to those presented there [4], which monotonously connects (v, θ 0 ) to (0, θ 1 = −δ). The magnitude of the chiral charge flux is controlled by that of θ 0 .

Conclusion and Discussions
In conclusion, assuming that the EWPT is of first order accompanying nucleation and constant growth of the broken-phase bubbles within the symmetric phase and that the dynamics of the bubbles are governed by the classical equations of motion of the gauge-Higgs sector, we have given some examples of the bubble profile by solving the equations of motion derived from the effective potential. According to the boundary conditions, the possible types of solutions are classified. Among them, the profiles of type (B), (C) and (D) will yield large CP violation around the bubble wall, though the type (D) seems hard to be realized dynamically. The solutions of type (B) require the parameters to admit spontaneous CP violation in some region between the broken and symmetric phases.
The explicit CP violation, which is small enough not in contradiction with observations, can avoid the cancellation of the generated baryon number between the CP -conjugate bubbles, leading to the BAU. The type (A) bubbles will yield insufficient BAU except for a restricted set of the carrier mass and the wall thickness.
If the temperature-dependent parameters are calculated in the MSSM at the oneloop level, the parameter f will be generated only by higher order terms in the hightemperature expansion of the squark or slepton contribution [9], and the condition dρ 2 − 4f ρ < 0 will be satisfied by d < 0. For d = v 2 /(4a 2 ) · λ 5 < 0 to be realized, we must have tree-level negative λ 5 in the two-Higgs-doublet model or radiatively induced negative λ 5 in the MSSM. In this case, we must avoid the problems peculiar to the spontaneous CP violation, that is, too light scalar and the domain wall. Both will be cured by a small explicit CP violation. In the MSSM, it will be unnatural to assume that all the soft supersymmetry breaking parameters do not introduce any explicit CP violation at the electroweak scale. As we noted in the previous section, if CP is spontaneously violated in the broken phase when δ = 0, a small δ nonperturbatively happens to drive large CP violation. This will further restrict some set of the parameters in the model. These constraints will be examined in a future publication. We have presented the chiral charge flux for the profile of type (A) and (B), which is calculated supposing that one of the Higgs doublets couple to a fermion mass eigenstate. In practice, the weak hypercharge plays its role and the hypercharge flux is additive for each fermion species in the standard model. This is because the up-type quark with positive hypercharge couples to e iθ 1 = e iθ/2 , while the down-type quarks or the leptons with negative hypercharge couples to e iθ 2 = e −iθ/2 or (e iθ 1 ) * = e −iθ/2 in the MSSM or the two-Higgs-doublet models with the discrete symmetry to avoid the tree-level FCNC. In a supersymmetric standard model, there also exist contributions from the superpartners with nonzero hypercharge such as the scalar partners of the fermions and those of the Higgs scalars. The effects of the chargino were considered in [16]. Finally, we comment on the discrete symmetry (2.21), which is a working hypothesis to reduce the dynamical degrees of freedom. The numerical method we employed to solve the boundary value problem is the relaxation algorithm, which varies the configuration to approach the correct solution starting from a prescribed initial configuration. Toward our final goal to find solutions to the equations for the full dynamical degrees without the discrete symmetry, the knowledge of the solutions obtained here will help us to prepare an appropriate initial condition for these equations. That is, as long as the solution with the discrete symmetry is stable in the enlarged functional space, we can obtain the desired solution by graduately modifying the parameters from those with the discrete symmetry to find the next initial configuration, step by step. In fact, we checked that for the solution of type (B), the discrete symmetry is not broken spontaneously, that is, the solution with the discrete symmetry is still a solution to the equations of motion for all possible order parameters.