Electroweak baryogenesis at high wall velocities

It is widely believed that electroweak baryogenesis should be suppressed in strong phase transitions with fast-moving bubble walls, but this effect has never been quantitatively studied. We rederive fluid equations describing transport of particle asymmetries near the bubble wall without making the small-wall-velocity approximation. We show that the suppression of the baryon asymmetry is a smooth function of the wall speed and that there is no special behavior when crossing the sound speed barrier. Electroweak baryogenesis can thus be efficient also with strong detonations, generically associated with models with observably large gravitational waves. We also make a systematic and critical comparison of our improved transport equations to another one commonly used in the literature, based on the VEV-insertion formalism.


INTRODUCTION
Electroweak symmetry exhibits a phase transition in the early universe, that is known to be a smooth crossover in the standard model (SM) 16 [1], but could become first order if new physics beyond the SM couples significantly to the Higgs boson. A strongly first order electroweak phase transition (EWPT) is one of the necessary requirements for electroweak baryogenesis (EWBG) [2][3][4][5], and it could also be a source of gravity waves that might be observed at LISA.
There is a perception that EWBG and observable gravity waves would tend to be mutually exclusive however, since the latter require very strong phase transitions, which lead to fast-moving bubbles, with wall velocity v w ∼ 1. This makes it difficult for particle asymmetries to diffuse efficiently in front of the wall and bias sphalerons to create the baryon asymmetry. There may be some tension between the two effects, but until recently there have been few quantitative studies including transport of the particle asymmetries [6][7][8][9][10][11]. Most works have focused on the coexistence of observable gravitational waves with the sphaleron washout condition v/T 1 [12][13][14][15][16][17][18][19][20][21][22], without taking into account the problem of reduced particle transport near the wall.
The theoretical deficit is in large part due to the fact that transport equations for the particle asymmetries have been derived using the approximation v w 1, making it impossible to reliably predict the baryon asymmetry at large v w . Since it was believed that v w 0.1 in the SM [23,24] and the minimal supersymmetric standard model (MSSM) [25], the small-v w approximation seemed adequate at the time. In recent years however, there has been increased interest in two-step phase transitions in- * Electronic address: jcline@physics.mcgill.ca † Electronic address: kimmo.kainulainen@jyu.fi volving a scalar singlet field [26], which is able to generate stronger phase transitions with typically higher v w . Such transitions can more easily satisfy the sphaleron washout constraint, and in addition can be a strong source of gravitational waves. It is therefore timely to revisit the transport equations relevant for EWBG and try to extend their applicability to higher v w . We will show in particular that nothing special happens when the wall speed crosses the sound barrier, and that the baryon asymmetry only vanishes smoothly in the extreme limit v w → 1. 1 This work has two main goals. The first is to update the fluid equations for the semiclassical force mechanism [27][28][29][30][31][32] to arbitrary wall velocities. This is strongly motivated because the currently existing formulation [33] breaks down for wall velocities exceeding the sound speed. Our second purpose is to perform a quantitative comparison between the semiclassical method and the competing "VEV-insertion" approximation [34,35]. These two approaches agree that the particle densities contributing to the baryon asymmetry are determined by (quantum) Boltzmann equations, but it remains controversial what precise form they should take.
The semiclassical method is designed to be valid when the de Broglie wavelength of the particles, of order the inverse temperature T −1 , is smaller than the typical width of a bubble wall L w . The interactions of particles with the wall can then be treated as coming from a semiclassical force, that can be derived using the WKB approximation [28,36,37], or from the closed-time-path (CTP) formalism of thermal field theory [29][30][31][32]. In the semiclassical approach the CP-violating force appears at the level of the Boltzmann equations. It is straightforward to approximate them by a set of moment equations with source terms induced by the force, that can be determined systematically in an expansion in powers of (L w T ) −1 (though the subleading corrections have not been computed).
The VEV-insertion method is also derived starting from the CTP formalism. Here quantum Boltzmann equations are manipulated to yield their classical counterparts at the level of integrated particle densities. In this approach the source term is not easy to extract and one must make a rather drastic approximation, expanding a two-point function to leading order in the spatially varying Higgs field VEV v(z) [34,35]. This is known as the VEV-insertion approximation. It can be regarded as an expansion in powers of v(z)/T , which cannot be very small inside the bubble if the phase transition is sufficiently strong to avoid washout. It is hoped that since v(z) is somewhat smaller inside the bubble wall, this can still be a reasonable approximation. But if that is the case, it must be capturing quite different physics from the WKB approach, since the two formalisms cannot be obviously reconciled, and in general they make quite different predictions.
For example, EWBG in the MSSM was analyzed using both formalisms [28,36,[38][39][40], with the VEV-insertion method giving significantly larger estimates for the asymmetry. However a systematic study of the differences between the two methods is lacking in the literature, in particular in comparing their predictions as a function of parameters characterizing the bubble wall. We will provide such a comparison in this work, for a prototypical model of CP violation in the wall. As was the case for studies of EWBG in the MSSM, we will demonstrate a large discrepancy between the predictions of the two methods.
We start in section 2 by arguing that the transport equations should not suffer from any sort of critical behvior for bubble walls that move near the speed of sound, but should rather only do so as v w → 1. In section 3 we review the derivation of the WKB transport equations and the origin of their v w dependence. We point out an inconsistency in the approximations used in ref. [33] (hereafter denoted FH06), and remedy it by a more careful evaluation of the coefficient functions for general values of v w . In section 4 we introduce our fiducial model and in section 5 we compare the predictions of the FH06 equations and our improved fluid equations in the semiclassical approach. In sect. 6 we quantitatively compare the semiclassical approach to the VEV insertion framework. Conclusions are given in sect. 7.

RELEVANCE OF WALL VELOCITY
The basic idea for reducing the full Boltzmann equations to a set of coupled first order fluid equations for the chemical potential and velocity perturbation, in the context of electroweak baryogenesis, was set out in ref. [27]. The method was elaborated for the MSSM in ref. [28] and for general two-Higgs doublet models in ref. [33]. In principle, one can always "integrate out" the velocity perturbation and convert the coupled system into a single second order diffusion equation for the chemical potentials, as was done in ref. [28]. However this is complicated by the fact that any particle that couples strongly to the Higgs boson (as required to source electroweak baryogenesis) has a mass that varies within the bubble wall, and therefore the coefficients in the diffusion equation are functions of z, the distance transverse to the wall. If one makes the crude approximation of ignoring baryon violation by sphalerons inside the bubble and taking the masses to vanish outside, then the z-dependence goes away and Green's functions techniques can be used to solve the diffusion equations. However this effectively approximates the wall as being very thin, which is inconsistent with the semiclassical expansion underlying the whole fluid approach. For quantitative results, one should numerically solve the coupled equations keeping track of the full z-dependence of the coefficients.
An important contribution of FH06 was to calculate all of the coefficient functions K FH i (x), where x = m(z)/T , appearing in the fluid equations. In their approach, these functions are independent of the wall velocity v w , which was achieved by expanding to leading order in v w . Therefore one could question to what extent this formalism can be accurate for walls with large wall velocities.
It is expected that diffusion lengths should diminish as v w gets large. One can make a simple estimate to quantify this statement, by asking what fraction F of particles in the plasma are moving faster than the wall, in its direction of motion. We would expect that only of order this fraction is able to contribute to the diffusion tail for the CP asymmetry in front of the wall. It is straightforward to show 2 that F = 1 2 ∞ γwvwm dp p 2 (1 − v w E/p)/(e βE + 1) ∞ 0 dp p 2 /(e βE + 1) for a massive fermion. In fig. 1 we plot F versus v w for particles with increasing values of m/T . It is clear that nothing dramatic happens near the sound speed v s ∼ = 1/ √ 3; instead v w = 1 is the only critical speed.

DERIVATION OF TRANSPORT EQUATIONS
The Boltzmann equation acting on the unperturbed distribution functions can be written in the wall frame as For a fermion with a CP-violating complex mass term m(z) = m(z)e iγ 5 θ(z) [28,30], where denotes ∂ z . Here E 2 wz = E 2 w − p 2 || and E w is the conserved wall frame energy. s k0 = 1 for particles and −1 for antiparticles, and s = ±1 for the states that are the eigenstates of the spin s in z-direction in the frame where the momentum of the state parallel to the wall p || vanishes. For the wall frame helicity eigenstates one should replace [41 where h = ±1 is the helicity and γ || = E w /E wz is the Lorentz boost for going to the frame where p || = 0. In practice the difference between the two spin bases is small [41]. In particular in the massless limit s h = h sign(p z ). Equation (3) is actually the definition of the physical momentum p z from the group velocity determined by the WKB dispersion relation [28,30]. It is convenient to write this relation in a form that defines E w in terms of the physical momentum: 3 We correct a typo in equation (E3) of [41], by replacing |pz| → pz.
where E ≡ p 2 + m 2 and E z ≡ p 2 z + m 2 . Using these variables, eqs. (3)(4) become.: Eqs. (6)(7)(8) agree with those derived in [33] when one sets s h → s. For bosons there is no CP-violating semiclassical force at this order in the gradient expansion [28]. However the CP-even kinetic force remains, and so all equations are valid for bosonic degrees of freedom if one simply sets s k0 = 0 everywhere.
The starting point for deriving fluid equations from the Boltzmann equation is to expand particle distribution functions around the equilibrium distribution. Because the kinetic momentum p z is conserved in collisions, the expansion in the rest frame of the bubble wall looks like where γ w = 1/ 1 − v 2 w , Here µ is a pseudochemical potential that defines the particle asymmetry and δf is an extra term whose specific form should be left unspecified, 4 except for stipulating that This condition is just the definition of µ; it ensures that δf does not affect the local particle density.

Classification by CP-parity
Next observe that the semiclassical force in (8) contains two distinct pieces: the first, CP-even term, is equal for particles and antiparticles while the second, CP-odd term, is opposite for particles and antiparticles. The CPeven term is of first order in gradients while the CP-odd term is of second order. Because of this hierarchy, one can solve the CP-even and CP-odd equations separately. To this end we introduce the definitions Using these together with eq. (6) we can write eq. (9) as where, expanding to leading consistent order in both CPeven and CP-odd quantities, where prime denotes d/d(γ w E) and The expansion (12) is also necessary for bosons. Even though bosonic equations do not have direct CP-violating sources at the order to which we are working, they can inherit CP-violating perturbations from their interactions with fermions.
To derive the CP-even equation we drop the CP-odd parts proportional to s k0 in the expansion (12) and in eqs. (7)(8) for the group velocity and the semiclassical force. After this the Boltzmann equation (2) immediately becomes where the Liouville operator is defined as and the CP-even source term is: The collision term for the CP-even perturbation δC e is model dependent and we do not specify it further until sect. 4. The CP-even equations (15)(16)(17) are valid both for bosons and fermions, and are helicity independent, unlike their CP-odd counterparts.

CP-odd equation
In the CP-odd sector we must account for the helicity. Because the relevant physical quantity for EWBG is the left-handed chiral asymmetry in front of the wall, one often concentrates only on the negative helicity sector, 5 but to be general we keep the full helicity dependence. Projecting out the CP-odd part of the Boltzmann equation (2) requires some work, but the final result is analogous to eq. (15) up to source and collision terms: where the CP-odd source term is (19) and the collision integral δC o is again model dependent, which we will specify later. Setting h = −1 and s p = sign(p z ), eq. (19) agrees with FH06 up to an overall sign. A number of CP-odd source terms computed in FH06, proportional to µ e and δf e , were dropped during evaluation, since they are higher order in gradients.

Moment expansion
One could solve µ e,o and δf e,o directly from Eqs. (15) and (18). It is more economical however, to first reduce them to a set of moment equations. Because of their identical forms, the equations for both CP parities can be treated simultaneously. We introduce moments by integrating over p, weighted by (p z /E) l , and dividing by a normalization factor where f 0,FD is the equilibrium distribution function for a massless fermion in the fluid frame. It is convenient to normalize even the equations for a massive particle using this universal factor, so that when several species of particles are coupled through their interactions, the rates in the collision terms are related in a simple way between equations for different species. Then terms appearing in the fluid equations can be expressed as averages over phase space of the form In particular the integrals over δf define the velocity perturbations The th moment of the evolution equation can then be written as Next we focus on the Liouville term (16), which contains important v w dependence.

Liouville term
Our goal is to reduce the system (23) to a closed set of equations for µ's and the velocity perturbations (22). We include only the two lowest moments, as has been done so far in the literature [33,41,42]. 6 Taking the zeroth and first moment of the Liouville operator we find where again denotes ∂ z except when acting on the distribution functions, where it denotes ∂ γwE , and we introduce the functions The D-and Q-functions are defined separately for bosons and for fermions, since the distribution function f 0w differs in the two cases. In the small v w -limit they reduce to the FH06-functions as Eq. (25) contains two problematic terms: u 2 is higher order in the expansion than the order to which we are working, and the last term is not obviously related to velocity perturbations (22). To treat the first term we need to introduce a truncation scheme, which relates u 2 to u 1 . (More generally one should relate the nth moment to moments u with < n.) Here we adopt a simple linear relation, henceforth denoting u 1 = u: u 2 ≡ Ru , where R is a function to be defined shortly.
To define the last term in (25) we need a further factorization assumption. Following ref. [28] and FH06, for any X that does not correspond to a velocity perturbation we replace where N 0 is another normalization factor, Unlike N 1 , N 0 is defined in terms of the massive distribution function f 0 of the particle under consideration.
In eq. (29), it may happen that X does not have any power of p z to be canceled by the factor E/p z in eq. (28). Nevertheless the integral can be defined using the Cauchy principal value. In particular After performing the singular angular integral using the principal value prescription, we find This should reduce to v wK FH 6 at leading order in v w , but due to a mistake in the evaluation ofK FH 6 in FH06 it does not.
Following FH06 we use the factorization rule also to define the truncation scheme, where the bracket average [·] is defined in (29). Then R becomes just the expectation value of the fluid velocity in the wall frame, which is an exact result.
The factorization and the truncation rules are of course somewhat arbitrary. It is therefore reassuring that thē R-term has but a weak effect on solutions: toggling between the choice (32) versus settingR ≡ 0 changes the final baryon asymmetry at the level of a few per cent. Moreover the definition (34) for R is reasonable because −v w is roughly the ratio between adjacent source terms, order by order in the moment expansion.

Sources and collision terms
Assembling the previous results, and including the collision and source terms, the fluid equations can be presented in full detail. Defining a vector w = (µ, u) T , the general form of the two moment equations may be expressed as where and S = (S 1 , S 2 ) T with S 1 = S and S 2 = (p z /E)S and similarly for the δC vector. The form (36) is generic to both CP-even and CP-odd sectors, which are only distinguished by their respective source terms.
Let us consider the source terms first. In the CP-even sector one finds using (17) with the definition In the small-v w limit one finds Q e 2 → K FH 3 . In the CP-odd sector, using eq. (19) similarly gives where the coefficient functions are with s p defined in (5). Setting s p → sign(p z ) one finds that Q 8o 2 → K FH 8 and Q 9o 2 → K FH 9 in the small v w -limit. Moreover, in previous work, the approximation S 1 = 0 was always made, because it is O(v 2 w ). For large velocities there is no hierarchy between S 1 and S 2 and one must include both sources.
It remains to consider the collision integrals. Both phase space averages δC 1 ≡ δC and δC 2 ≡ (p z /E)δC are normalized using N −1 1 , eq. (20). The collision term moments are derived following appendix A of ref. [28], Here s ij = 1 (−1) if the corresponding species is in the initial (final) state in the interaction with rate Γ i , and Γ tot = i Γ i is the total interaction rate, including elastic channels that do not contribute to the sum in δC . The normalization factor was neglected in FH06. For a massless fermion, for example, K 0 ∼ = 1.1, Eqs. (42) are valid for both the CP-even and the CP-odd cases. Eqs. (35) obviously depend on a large number of coefficient functions: D , Q , R,R, Q e , Q 8o , Q 9o and K 0 . Most of these depend on the wall velocity v w and the dimensionless ratio x = m/T . However they are universal and model independent. In practice, we compute them on a grid of x and v w values and spline fit them. Explicit expressions are given for all the integrals definng them in appendix A.

Critical speed predictions
There is a widespread notion, apparently originating from ref. [44], that diffusion is inefficient for wall speeds exceeding the plasma sound speed. This would mean in particular that EWBG would not be feasible for detonation walls, corresponding to very strong phase transitions, often invoked in the context of gravitational wave production. This assertion is not true, as we shall show, but it turns out that the FH06 equations are, quite fortuitously, consistent with the false assumption.
We noted that the fluid equations can be written in the matrix form Aw = F [w], where A is given in Eq. (36), while in FH06 the A matrix is settingK FH 5 = −1 as mentioned above. One can solve for the value of v w where A becomes singular (noninvertible) using det(A) = 0. If a solution exists for v w < 1 it implies a critical speed v c beyond which diffusion is quenched. The exact prediction using the A-matrix in (36) gives whereas the approximate FH06-condition gives a different velocity The dependence of v c on m/T as obtained in the FH06case (46) is shown in fig. 2 for a Fermi-Dirac distribution.
(The corresponding curve for bosons look similar.) For light particles the quench limit is maximal and very close to the sound speed, but this is a mere coincidence due inappropriate use of the small v w -approximation. Indeed, from (45), employing full v w -dependent function, we find that v c = 1, in accordance with the arguments given in section 2. Thus diffusion efficiency should go to zero smoothly as v w → 1, with no particular features at the sound speed, v w = v s . We will show that this indeed is the case.

PHENOMENOLOGICAL MODEL
To illustrate the consequences of our improved transport equations, we will compute the baryon asymmetry that they predict in a prototypical model that gives rise to EWBG, where the top quark mass has a z-dependent CP-violating phase in the bubble wall. The mass term can be written as in terms of the chiral components, where m t = y t v(z) and v(z) is the Higgs VEV that varies spatially within the wall. It can occur in two-Higgs-doublet models, or in singlet plus doublet models where a dimension-5 operator like i(s/Λ)Q 3 Ht R contributes a phase to the top mass, if s also gets a VEV in the bubble wall. In such a model, the effective top quark mass term takes the form which implies Here we will not consider the CP-even equations, which would be relevant for computing the wall speed and shape. Instead, we concentrate on the CP-odd sector and take a phenomenological approach, where v w is treated as a free parameter, and the VEVs h(z), s(z) are modeled as We are primarily interested in the v w -dependence of the results and therefore choose fiducial values for the other parameters, v n = 1 2 w n = T n , Λ = 1 TeV in terms of the nucleation temperature, taken to be T n = 100 GeV.

Fluid equation network
With the tools and notation developed in the previous section one can express the complicated equation network in a compact form. Our system consists of four particle species 7 strongly coupled by the top-Yukawa interactions: left and right helicity tops are respectively denoted by t L and t R , left helicity bottom by b L and the complex Higgs particle by h. We neglect the small difference between helicity and chirality of the fermions here. There are eight dependent variables, combined into four 2-vectors where the A− and B-matrices are defined in (36) and the sources are S i = (S i1 , S i2 ) T , S i given by equation (39) taking h ≡ −1 for the left-handed fermions. In practice S b is neglible due to the smallness of the bottom Yukawa coupling.
In addition to top-Yukawa interactions, we account for the W boson interactions that tend to equalize the t L and b L chemical potentials, the strong sphalerons, top mass insertions (helicity flips) that damp the combination µ tL − µ tR (µ h ), and Higgs damping from electroweak symmetry breaking. These are the same collision terms as in FH06. and Explicit equations for light quarks are not needed, even though their chemical potentials appear in the strong sphaleron rateΓ SS [µ i ] = Γ SS q (µ qL − µ qR ), since their chemical potentials can be determined analytically. Light quarks are activated only by strong sphalerons, and in the approximation of 7 We differ from the notation of FH06 by keeping track of the asymmetry in the right helicity sector with t R instead of its conjugate t c R . The CP-odd asymmetries of the two species have a relative sign, µt R = −µ t c R , since each CP-odd variable represents the difference between the two C-conjugate species. Likewise, the source for µt R corresponds to h = 1, which is opposite to that for µ t c R .
no Yukawa mixing, µ qR = −µ qL for all light species. Then using baryon number conservation (neglecting electroweak sphalerons, which are slow on the relevant time scale), B = q (n q −n q ) = 0, one finds where D 0 = f 0w is a special case of the function (26) with = 0, identical to K FH 1 for all v w , as can be shown by partial integration. Using (54) the strong sphaleron rate can be written as (55) Inelastic collisions induce mixing between the particle species. Eqs. (52) are nevertheless linear in µ i and u i , and can be written in the compact matrix form is tridiagonal, and the matrix Γ combines the m 2 i B i and collision terms. The source vector is (52). Because of its block structure, A is easily inverted to yield This system is best solved using relaxation methods [45] since shooting tends to be unstable. The 8 × 8-matrix A −1 Γ is the Jacobian of the differential equation network and its eigenvalues' signs distinguish the growing and decaying modes at the boundaries. This information may help to improve the numerical stability in more complicated systems. Once the chemical potentials for the perturbations are determined, the baryon asymmetry follows from integrating them in the sphaleron rate equation. Following for example ref. [46] (but including the full Lorentz-covariant relations) we find: 8 The seed asymmetry in eq. (58) is the chemical potential for left-handed baryon number, µ BL = 1 2 q µ qL , which can be written in terms of µ tL , µ bL and µ tR using baryon number conservation: The function f sph (z) = min(1, 2.4 Γ sph T e −40h(z)/T ) is designed to smoothly interpolate between the sphaleron rates in the broken and unbroken phases. g * is the number of degrees of freedom in the heat bath; in the standard model g * = 106.75. 8 To our knowledge, the 1/γw factor in front has been omitted in previous literature. It arises from the change of variable dt → dz/(vwγw) with z in the rest frame of the wall.

COMPARISON TO FH06
We can now compare our improved fluid equations (52)(53) to those of FH06. The only difference between the two lies in the definition of the various coefficient functions, which we have renamed at the same time correcting and generalizing them to arbitrary wall velocities. To facilitate the comparison the results are collected in a dictionary translating between the two naming schemes in table I. Our equations agree with those of FH06, when one assumes h → −1 and s p → sign(p z ) in the sources and replaces the coefficent functions according to table I.

CK
FH06 For the interaction rates we use the values given in [42]: where top mass is as given in (48) and m 2 W ≡ g 2 h(z) 2 /4. Furthermore the total interaction rates were defined as Γ i tot = K FH 4,i /(D i K FH 1,i ) with a quark diffusion constant D q = 6/T and a Higgs diffusion constant D h = 20/T . The numerical impact of the Higgs and bottom masses is found to be quite small, and following FH06 we take them to be massless. Many of these rates have been quite roughly estimated, in some cases going back to the early reference [44], and deserve to be updated. We hope to make better determinations in an upcoming paper.
We display dependences of the predicted baryon asymmetry of the universe normalized to the observed value, BAU ≡ η B /η B,obs , in fig. 3. In both panels the thick red solid lines labeled "CK-s" correspond to the improved fluid equations with the spin-s source, where we set s p → sign(p z ) and h = −1 in eqs. (39)(40)(41). The thick dash-dotted blue lines labeled "FH-s" correspond to the same spin-s source, but using the FH06 equations 9 . Thin dashed green lines labeled "CK-s, K 0 = 0" correspond to the case where we set the K 0 -function to unity in the otherwise accurate equations. The thin black dashed lines labeled "CK-h" correspond to the improved fluid equations with the helicity source, still taking h = −1, but with s p given by eq. (5). Clearly all approximations agree very well for v w 1 as expected, since the two sets of functions largely agree in the small v w -limit; for v w < ∼ 0.01, the only significant difference between the CK and the FH06 solutions comes from K 0 . For larger v w the predictions differ sig- nificantly, in accord with our general arguments. In particular, the FH prediction plummets as v w approaches the sound speed v s = 1/ √ 3, shown by the thin vertical line in the plots. The more exact treatment on the other hand does nothing special near v w = v s ; as expected the BAU smoothly decays as v w → 1.
Using the spin-s source corresponds to identifying chirality with the eigenstates of spin in the z-direction, in the frame where the parallel momentum of the state vanishes, whereas the helicity source identifies chirality with helicity. The difference between the two is found to be small, due to the two bases becoming degenerate in the massless limit; in our example all fermions are massless in front of the wall.
In figure 4 we show the dependence on L w with v w = 0.5 held fixed. The FH prediction is substantially higher than the accurate value and its ratio to the correct solution remains nearly constant. To summarize, our results and those of FH agree reasonably well for small v w , but the improved fluid equations should be used for v w > ∼ 0.1 to get accurate results, and for v w > ∼ v s they are essential, since the FH equations incorrectly predict a vanishing BAU.

COMPARISON TO OTHER FORMALISMS
There has been a long-standing divide among practitioners of electroweak baryogenesis as to which transport equations to use; yet no systematic comparison between them has been made in the literature. We undertake to do so in this section, continuing with the ansatz for the wall profiles and spatially varying top quark mass (48,50) introduced previously.
The VEV insertion formalism is derived at the level of the integrated particle densities, which is equivalent to the formalism introduced in refs. [44,47], consisting of coupled second-order diffusion equations for the local particle densities, in matrix form, Here µ = (µ tL , µ bL , µ tR , µ h ) T and S = (S t , 0, −S t , 0) T , D = diag(D q , D q , D q , D h ) is a diagonal matrix of diffusion coefficients and δC[µ] is the inelastic collision integrals. The unsourced equation for µ bL is usually omitted in the literature [48,49]. In this case baryon number conservation (as discussed above) leads to conditions µ qL = µ qR and µ bL = −(D t 0 /D b 0 )(µ tL + µ tR ) [44,48]. In fact this is a reasonable approximation, as one can see from figure 5, where we plot the chemical potentials for our fiducial case (51) and for the spin-h source using our improved fluid equations. For the purpose of comparing the formalisms however, we have included the b L degrees of freedom in the diffusion equation networks.
In order to compare the system (60) to our improved equations (52), we need to find the equivalent source term to use in (60). The standard way to do this [28] is by eliminating u from the WKB equations (35), neglecting all (m 2 ) µ and (m 2 ) u terms in the derivation. It is easy to show that this procedure yields the following results adequate for the WKB-picture 10 11 10 One should not confuse the diffusion coefficients with the fluid equation coefficient functions D i . The latter are distinguished from the former by the fact that they are always associated with a numeral index. 11 Note that the numerator in the diffusion coefficient D WKB is just the determinant of the matrix A in fluid equations (35). Thus, the critical speed condition discussed in section 3.6 corresponds to zero diffusion length. For vw > vc the diffusion length would be negative, which is of course unphysical. This is why the FH06 solutions go to zero as vw → vc. However, in the improved where δC[µ] terms are given in (53). In the small v w -limit D α WKB → D α and S α WKB → D α S α,2 / v 2 z for each species α = q, h, in agreement with ref. [28].
The VEV-insertion formalism predicts a very different form for the source term in (60). We give a detailed derivation in appendix B. The result, normalized as in eq. (60), is where N c = 3 is the number of colors of the top quark and I ∼ = 0.4 is an integral given in appendix B.
As for the diffusion term, no dependence of D on v w is considered in the earlier literature in the VEV insertion approach. Accordingly, we will use As mentioned above, we employ the same equation network for the WKB-and the VEV insertion mechanisms, so that slightly upgrading the network of ref. [49] to include b L , we set where as before δC[µ] correspons to eq. (53).
We can now compare the semiclassical and VEVinsertion formalism predictions for the BAU on a level playing ground, using the diffusion equations (60). The results are shown in figs. 6. The upper panel displays the absolute value of the BAU ≡ |η B /η B,obs | as a function of the wall velocity for our fiducial model (51) and for the spin-h source. The thick solid black line corresponds to the solution of the diffusion equation (60) with the WKBvariables (61). The dashed red line shows for comparison the result of our earlier calculation with improved fluid equations. The two semiclassical approximations agree remarkably well, in particular for large v w .
Considering the VEV-insertion formalism, the thick dash-dotted blue line (labeled "VEV-VEV") displays the result of using the parameters (62-64) in eq. (60). The VEV-insertion method predicts 10 − 50 times larger asymmetry than does the semiclassical method. The thin dashed green line (labeled "VEV-WKB") shows the result of using the VEV-insertion source, but WKB diffusion constant and collision terms from eq.  v w the difference between the VEV-insertion method and the WKB-method is dominated by the source terms. However, for v w > ∼ 0.4 a further deviation is caused by the v w -dependence of the evolution equations. The lower panel displays the dependence of BAU on the wall width L w for v w = 0.5. For L w T > ∼ 2 the two semiclassical approximations agree very well. The difference at small L w is expected, because the diffusion approximation neglects several m 2 -corrections included in the fluid equations. As the wall width increases, the semiclassical BAU decreases rapidly. This is the expected behaviour as the system becomes increasingly classical. The VEV-insertion predicts a much slower decrease of BAU with L w .
In our opinion, the semiclassical formalism is the more reliable one. First, it has a clearly established small expansion parameter: (L w T ) −1 , and the results show the expected behaviour as a function of this parameter. Second, it has been derived both using WKBmethods [28,36,37] and from the fundamental CTPformalism [29][30][31][32]. Third, the semiclassical limit has been recently shown to arise in a full quantum mechanical treatment without any gradient expansion [50] (for earlier related work see [51] and [52][53][54][55][56][57][58].).
In the VEV-insertion approach the expansion parameter is not as clearly defined: in addition to expanding in powers of m(z)/T , there is also a gradient expansion, which however always results in fewer derivatives acting on the source term than in the semiclassical approach, leading to the different dependence on the wall width shown in fig. 6. Moreover, the source term is infrared singular, cured only by introducing a damping term that is not related to the quantum physics of the problem. This can be seen in eq. (B1), which blows up when the damping rate γ → 0. The lack of convergence of the expansion in m/T was recently investigated in ref. [59], where it was shown that higher order terms are only small if m/T < √ α, where α is the relevant interaction strength of the fermion of mass m. There is a claim in the literature that the m/T expansion can be avoided [60], but this is based upon a phenomenological approach which, although superficially similar to the VEV-insertion formalism, does not derive the transport equations from first principles within the CTP formalism.

CONCLUSIONS
In this paper we generalized the semiclassical fluid equations for electroweak baryogenesis to the regime of arbitrarily large wall velocities, showing that diffusion remains relatively efficient for v w exceeding the sound speed in the plasma. As a result, EWBG can be effective even for very strong transitions corresponding to detonations. We performed a detailed comparison between a new improved network of fluid equations and the previous formulation by Fromme and Huber [33]. For small wall velocities v w < ∼ 0.1 the two formulations agree reasonably well, but for larger values, in particular for v w exceeding the sound speed, the improved formalism is indispensible.
We then quantitatively compared the semiclassical fluid equations to a competing framework, the VEV insertion method. To do so required reducing the semiclassical fluid equations, consisting of coupled equations for chemical potentials and velocity perturbations, to a set of (WKB) diffusion equations for the particle densities alone (while retaining the full velocity dependence). This was necessary because the VEV-insertion formalism is derived at the level of particle densities, and does not lend itself to a more accurate approximation of the Boltzmann equation in terms of coupled moments (including the velocity perturbation).
The WKB diffusion equations agree very well with the improved fluid equations in the semiclassial picture. The VEV insertion method on the other hand predicts the BAU is a factor of 10-50 times larger than in the semiclassical method. This difference arises mostly (especially for small v w ) from the different source terms in the semiclassical and the VEV-insertion schemes. We argued that the semiclassical results are more reliable, as they have been derived and verified in various different approaches and they, unlike the VEV-insertion results, show the expected parametric behaviours. Finally, we caution that while our improved fluid equations cure the incorrect dependence predicted by the FH06 network at large v w , they still correspond to a loworder expansion in moments of the particle distribution functions f . One might reasonably expect that perturbations δf could be highly nonGaussian, such that going beyond second order in the moment expansion could significantly modify the results presented here. This is a question that deserves further study.
an error in that equation [62]. The correct expression is where the superscripts p, h refer to quasiparticle and hole excitations in the hot plasma, taken to be in the electroweak symmetric phase and n F denotes the Fermi-Dirac distribution function evaluated at a complex energy E = E + iγ, where γ is the damping rate (thermal width) of the left-or right-handed top quark. The error in the original expression for (B1) was that the term "0" was originally "1", which leads to a UVdivergent integral. It is argued [63] that normal ordering removes this term, although no derivation has ever been published. The correction to a similar source term was mentioned in ref. [64] (see [31] of that paper).
The VEV-insertion source has the peculiar property of being singular if the energies E are real. The regulating damping rate is dominated by the QCD contribution [65]: The real parts of the energies are given approximately by (these are good analytic fits to the numerical solutions for the poles of the thermally corrected propagators) with m being the thermal mass for the chirality of interest [66], m 2 L = g 2 s /6 + 3g 2 2 /32 + y 2 t /16 T 2 m 2 R = g 2 s /6 + y 2 t /8 (B4) The wave-function normalization factors are given by for each kind of particle or hole. Taking the known values of the coupling constants, the integral in (B1) then becomes I ∼ = 0.4 T 3 . It is normalized in this form to be a source for the diffusion equation for particle densities. To convert it to a source for the top quark chemical potential, we use the relation δn = N c g t D 0 T 2 µ/6 for a chiral quark with g t = 2 spin degrees of freedom. Hence S t = S VEV /(D 0 T 2 ) to obtain eq. (62).