1 Introduction

Nikhef-2017-044

Electroweak Baryogenesis and the

Standard Model Effective Field Theory

Jordy de Vries, Marieke Postma, Jorinde van de Vis, Graham White

Nikhef, Theory Group, Science Park 105, 1098 XG, Amsterdam, The Netherlands

ARC Centre of Excellence for Particle Physics at the Terascale School of Physics and Astronomy, Monash University, Victoria 3800, Australia

TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia V6T 2A3, Canada

We investigate electroweak baryogenesis within the framework of the Standard Model Effective Field Theory. The Standard Model Lagrangian is supplemented by dimension-six operators that facilitate a strong first-order electroweak phase transition and provide sufficient CP violation. Two explicit scenarios are studied that are related via the classical equations of motion and are therefore identical at leading order in the effective field theory expansion. We demonstrate that formally higher-order dimension-eight corrections lead to large modifications of the matter-antimatter asymmetry. The effective field theory expansion breaks down in the modified Higgs sector due to the requirement of a first-order phase transition. We investigate the source of the breakdown in detail and show how it is transferred to the CP-violating sector. We briefly discuss possible modifications of the effective field theory framework.

1 Introduction

The asymmetry between baryons and anti-baryons, characterized by the ratio of densities of baryon number and entropy, has been determined by two independent methods [1, 2]

(1)

which are in good agreement. The nonzero value of provides one of the strongest indications that the Standard Model (SM) of particle physics is incomplete. While the SM has a sufficiently rich structure to in principle fulfill the three Sakharov conditions [3], in practice it gives rise to an asymmetry that is too small by many orders of magnitude. The first problem is that the electroweak phase transition (EWPT) is a cross-over transition, whereas the required strong first-order transition can only occur for a much lighter Higgs boson than is observed [4, 5, 6, 7]. The second problem is that the amount of CP violation in the SM is not sufficient to produce the observed baryon asymmetry [8, 9, 10].

Understanding why baryons are more abundant than anti-baryons thus requires beyond-the-SM (BSM) physics. Such BSM physics could live at a very high energy scale, decoupled from the electroweak scale, as occurs for instance in (most) scenarios of leptogenesis. Such scenarios, while well motivated, will be difficult to probe in current and upcoming experiments although measurements of neutrinoless double beta decay would point towards them. In scenarios of electroweak baryogenesis (EWBG) [11, 12, 13], however, the scale of BSM physics cannot be much higher than the electroweak scale which makes the scenario more testable. In particular, searches for new scalars, precision measurements of Higgs couplings, and electric dipole moment (EDM) experiments all probe different aspects of EWBG scenarios.

The above considerations have led to a large number of SM extensions that can lead to successful EWBG. Depending on the BSM details, such as the particle content and symmetries, different tests are required and each scenario requires a detailed phenomenological study. It would be a great advantage if the crucial aspects of all these models can be tested in a single framework. In principle, the SM Effective Field Theory (SM-EFT) could provide such a framework [14, 15, 16, 17, 18, 19, 20, 21] as it provides a model-independent parametrization of BSM physics. The SM-EFT assumes that any BSM degrees of freedom are sufficiently heavy, such that they can be integrated out and that their low-energy effects can be captured by effective gauge-invariant operators containing just SM degrees of freedom. While an infinite number of effective operators exist, they can be organized by their dimension. The higher the dimension of the operators, the more suppressed their low-energy effects are by powers of , where is a typical low-energy scale, such as the electroweak scale, and the scale of BSM physics. The first operators relevant for EWBG appear at dimension-six. If the SM-EFT is suitable for the description of EWBG, it would provide an attractive framework as the dimension-six operators have to a large extent been connected to low- and high-energy experiments already, while the EFT operators can be easily matched to specific UV-complete models.

The applicability of the SM-EFT requires a perturbative expansion in , which is potentially dangerous for EWBG applications. Extending the SM scalar potential with a dimension-six cubic interaction to ensure a strong first-order EWPT requires a relatively low scale GeV [14], which can lead to a mismatch between calculations in the SM-EFT and specific UV-complete models, see for instance Ref. [19] for an analysis of the singlet-extended SM. Furthermore, EDM constraints on dimension-six CP-violating (CPV) operators potentially relevant for EWBG are typically in the multi-TeV range [22, 23, 24, 25]. This difference in scale can be accommodated by assuming a different threshold for the CPV dimension-six operators such that [16]. In this way, it might be possible to use EFT techniques for the CPV sector despite the relatively low scale required for a strong first-order EWPT.

In this work we investigate a related issue of the EFT approach to EWBG. As mentioned, the EFT approach requires that the effects of higher-dimensional operators are suppressed with respect to the lower-dimensional ones. For energies () around the electroweak scale and GeV, the expansion parameter seems at first sight to be sufficiently small for a perturbative expansion. In practice, the necessity of a first-order phase transition requires a fine balance between dimension-two, -four, and -six contributions to the Higgs potential. While no such balance is necessary for the CP-violating sector, successful EWBG requires an interplay of the scalar and CPV sectors, such that formally higher-order corrections to the latter might become relevant as well. To study this, we consider two specific EFTs which can be related via the classical equations of motion (EOMs). EOMs can be applied to EFTs to reduce the number of operators in the EFT basis [26]. Operators related via EOMs lead to identical observables up to higher-order corrections in the EFT expansion. That is, if the EFT is working satisfactory the two EFTs under investigation should lead to the same baryon asymmetry modulo small corrections. The main goal of our work is to perform a detailed test of this hypothesis.

A somewhat similar study was performed in Ref. [21], where it was concluded that the derivative operators in the EFT can no longer be eliminated by EOMs without explicitly specifying the dynamics of the phase transition. We improve on these results by carefully investigating — both analytically and numerically — the redundancy of the operators in the EFT, including important thermal effects. We also improve the EDM phenomenology with respect to Ref. [21], which neglected several relevant contributions.

Our study allows us to pin down where and how the EFT approach breaks down for the application of EWBG. We find that scenarios that are identical up to higher-order dimension-eight corrections lead to large differences in the baryon asymmetry. The breakdown of the EFT is not specific to the EWBG calculation and in principle also arises at zero temperature where certain CPV interactions get -corrections from dimension-eight operators. However, these interactions are largely unconstrained, and as far as the EDM phenomenology is concerned the scenarios that are related by the EOMs are equivalent. In the context of EWBG, however, we find that dimension-eight corrections strongly modify the strength of the CPV source term that drives the creation of the matter-antimatter asymmetry. While this modification is partially washed out due to SM processes that are active during the phase transition, it still leads to a reduction of the matter-antimatter asymmetry by a factor . Higher-dimensional CPV operators can therefore not be neglected.

Our paper is organized as follows. In Sect. 2 we introduce the SM-EFT operators we consider and how they are related via the EOMs. We also obtain the EDM constraints on the CPV operators. In Sect. 3 we discuss details of the EWPT. In Sect. 4 we review the derivation of the transport equations that describe the plasma in front of the bubble walls. We focus on how the source term that drives the asymmetry depends on the CPV operators. It is important to take thermal corrections to the CPV operators into account and these are calculated in Appendix A. The baryon asymmetry is calculated in Sect. 5. Most details of the solution of the transport equations and the values of the parameters that are used in the computation are delegated to Appendix B. With the calculated asymmetries we test the impact of formally higher-order corrections, and identify the source for the breakdown of the EFT expansion. We summarize, conclude, and give an outlook in Sect. 6.

2 Effective scenarios for electroweak baryogenesis

We begin by defining the SM Lagrangian. We write the Lagrangian in terms of left-handed quark and lepton doublets, , and, , respectively, and right-handed singlets , , and . The field represents the Higgs doublet of scalar fields . We define , where is the antisymmetric tensor in two dimensions (). The covariant derivative is given by

(2)

where , , and are, respectively, the , , and coupling constants. and denote and generators, in the representation of the field on which the derivative acts. The hypercharge assignments, , are , , , , , and for , , , , , and , respectively. The field strengths are

(3)
(4)
(5)

with and denoting the and structure constants. The SM Lagrangian is then written as

(6)

We have suppressed fermion generation indices, but note that the Yukawa matrices are general matrices in flavor space. In this work, we are mainly interested in interactions of the third generation of quarks. We neglect the Yukawa couplings to light fermions, but make an exception for the electron Yukawa which plays an important role when considering EDM constraints. We have left out the topological theta terms which play no role in our discussion.

The full set of dimension-six gauge-invariant operators was constructed in Ref. [27] and updated in Ref. [28]. There exist a large set of operators but only relatively few have impact on EWBG [18, 21]. Here, we consider two specific scenarios, which we label by scenario A and B, in which we consider a small subset of dimension-six operators:

  1. Here we extend the SM Lagrangian by two dimension-six operators

    (7)

    where and are dimension-six couplings. and denote, respectively, the left-handed doublet of the third-generation quarks and the -component of the up-type Yukawa-coupling matrix. The first term in Eq. (7) modifies the scalar potential and will be used to ensure a strong first-order EWPT. The second term is a dimension-six modification of the top Yukawa coupling which causes a misalignment between the top-quark mass and the top-Higgs coupling such that the latter can obtain a physical CPV phase. In fact, for simplicity we consider a purely imaginary coupling , with . This particular choice of dimension-six operators has been well studied [15, 16, 20, 21] and is sometimes called the minimal EWBG scenario [16].

  2. In this scenario we add the same modification to the scalar potential, but consider a different CPV structure. We use

    (8)

    where and denote, respectively, the lepton doublet of the first generation and the real electron Yukawa coupling. is a real constant introduced for normalization purposes. The second term provides the dimension-six CPV source for EWBG, while the third term describes a CPV top-electron coupling and is introduced for later convenience. As in scenario A we consider a purely imaginary coupling , with .

It is possible to relate the two scenarios via the classical EOM for the scalar field [26]. From the Euler-Lagrange equations we obtain

(9)

where we neglected the Yukawa couplings to other fermions and a term proportional to . Applying the EOM to Eq. (8) shifts the Lagrangian into111Here we used that , for purely imaginary .

(10)

where the top-electron term in Eq. (8) has cancelled and the dimension-eight piece is given by

(11)

which scales as  . If the EFT is working satisfactory this term should give rise to small corrections compared to the dimension-six terms in Eq. (10). It is possible to simplify Eq. (10) by redefining the and in order to absorb the term into the SM top-Yukawa coupling. The resulting Lagrangian then becomes

(12)

which is of the same form as Eq. (7) modulo the higher-order correction. For now, we will not remove the piece and keep the form of Eq. (10), mainly because it provides a cleaner relation between and the derivative of the scalar potential.

2.1 Zero-temperature phenomenology

We now discuss experimental constraints on the dimension-six Lagrangians. We begin with the Lagrangian in scenario A. We assume the scalar field picks up a vacuum expectation value (vev) GeV, and work in this section in the unitarity gauge , where denotes the Higgs boson with zero-temperature mass GeV. Because of the modified scalar potential, in both scenarios the relations between the parameters and on the one hand and and on the other, are modified by the term. At zero temperature we can express

(13)

Effects of the dimension-six interaction in particular induce deviations of the Higgs cubic and quartic interactions with respect to SM predictions. This manifests in processes such as double Higgs production, see e.g. Refs. [29, 30] for recent discussions. At the moment, such processes have not been accurately measured and current constraints on are weak.

In scenario A, the dimension-six term in Eq. (7) gives a contribution to the top mass. We define the real top mass by

(14)

Although this relation implies that obtains a small imaginary part , this imaginary part only enters observables at which can be neglected. As such, from now on we use . The interactions between top quarks and Higgs bosons become

(15)
Figure 1: Two-loop diagrams contributing to the electron EDM. Single (double) lines denote the electrons (top quarks), dashed lines the Higgs boson, and wavy single (doubles) lines the photons (Z-bosons). Circles denote SM vertices, while squared denotes CPV dimension-six vertices. Only one topology for each diagram is shown.

The top-Higgs interactions pick up a CPV component which can be probed in EDM experiments. In particular, the strongest constraint comes from the ACME experiment using the polar molecule ThO, which sets a strong limit on the electron EDM222This limit assumes negligible contributions to the ThO observable from CPV semi-leptonic operators. This is justified in our scenarios as these semi-leptonic operators are only induced at loop level and strongly suppressed by small Yukawa couplings. at c.l. [31]. The dominant contribution to the electron EDM from the CPV top-Higgs couplings arises from the two-loop Barr-Zee diagram333We neglect diagrams where the internal photon is replaced by a -boson. These are suppressed by the electron-Z vector coupling , where is the square of the sine of the Weinberg angle. in Fig. 1a [32] and is given by

(16)

in terms of the number of colors , the electron mass , , and the two-loop function

(17)

The electron EDM limit then sets the strong constraint . If we assume , we obtain TeV.

In scenario B, the analysis is slightly more complicated. After electroweak symmetry breaking and assuming a purely imaginary , the CPV operators relevant for the EDM calculation become

(18)

in terms of the Z-boson mass, , and the dots denote interactions with two or more gauge bosons, which play no role in the EDM calculation. The last two terms in Eq. (18) contribute to diagrams 1b and 1c and mutually cancel (this was the reason to include the CPV top-electron coupling in Eq. (8)). The first two terms contribute to diagrams 1a and 1b. The contributions can be combined by using inside the loop, and together become

(19)

which is of the same form as Eq. (16), but with the replacement . By specifying , we can ensure the same electron EDM predictions in the two scenarios. In what follows below, we will use

(20)

with the constraint TeV from the limit on the electron EDM.

In Sect. 2 we argued that scenario A and B are the same apart from higher-order corrections. So where are these higher-order corrections in the EDM calculation? To answer this question it is useful to look at Eq. (10), which is the CPV Lagrangian after applying the EOM to scenario B. The physical real top mass is now given by

(21)

where the last equality follows from Eq. (13). After setting to its value in Eq. (20), the interactions between top quarks and Higgs bosons become

(22)

where the dots denote terms with four and five Higgs bosons. Comparing this to Eq. (15), we see that the dimension-eight corrections, , only affect interactions with two or more Higgs bosons. These terms only contribute to the electron EDM at three loops and these contributions are therefore strongly suppressed. As such, as far the EDM phenomenology is concerned, scenarios A and B are essentially identical.

The CPV top-Higgs interactions give rise to the EDMs and chromo-EDMs of light quarks via very similar Barr-Zee diagrams. Another two-loop diagram involving a Higgs exchange inside a closed top-loop connected to external gluons, gives rise to a CPV three-gluon operator, the so-called Weinberg operator [33]. The quark (chromo-)EDMs and Weinberg operator in turn give rise to EDMs of the neutron and diamagnetic atoms such as Hg and Ra. With current experimental sensitivities, these limits are not competitive with the limit from the electron EDM. Furthermore, the hadronic and nuclear EDMs are sensitive to theoretical uncertainties due to hadronic and nuclear matrix elements. A much more detailed discussion can be found in Refs. [23, 34].

Finally, the CPV top-Higgs coupling can be directly probed in collider experiments, see e.g. Refs. [35, 36, 37, 38]. However, for the foreseeable future, the resulting limits are significantly weaker than EDM constraints [34].

3 The electroweak phase transition

3.1 The finite-temperature Higgs potential

For the measured value of the Higgs mass, the EWPT in the SM is a cross-over such that the Sakharov condition demanding an out-of-equilibrium process is not satisfied [39, 40, 6]. We have supplemented the Higgs potential in both scenarios therefore by an effective dimension-six operator. In this section we work in the Landau gauge and define the components of the Higgs field as

(23)

with the Goldstone bosons, the Higgs field, and the background field, the tree-level classical potential in terms of is given by

(24)

In order to describe the phase transition we need to include loop corrections to the potential. The one-loop effective potential can be split into the zero-temperature Coleman-Weinberg potential and the finite-temperature contribution. The former can be resummed to get the renormalization group improved effective potential where the couplings are running with scale. For the analysis of EWBG we use the coupling values at the renormalization scale , and for simplicitly neglect all running effects and threshold corrections. The calculation of the finite temperature contribution is reviewed in Appendix A. We can then write the one-loop effective potential as , with the renormalization-group (RG) improved potential, and

(25)

The sums are over all bosons respectively fermions that couple to the Higgs. We only include the fermion contribution from the top quark. and denote the degrees of freedom and are given by , with the number of colors. The functions are given by

(26)

with the upper (lower) sign for bosons (fermions). In the high-temperature expansion (see Eq. (A.11) for the expansion of and ) the potential becomes

(27)

where

(28)

with and the zero-temperature Higgs mass and Higgs vev, respectively. For simplicity, we will use this high-temperature expansion to determine the allowed values of , and to find the Higgs profile accros the bubble wall that is used for the calculation of the baryon asymmetry. In addition, we neglect higher-loop corrections due to ring diagrams (usually called daisy resummation), and evaluate all running couplings at the scale of the Z-boson mass, and as mentioned above neglect further running effects and threshold effects. The results are not significantly different from those obtained with the full potential [17], in which all these effects are included. Keeping in mind the main goal of this work – to compare EWBG in the two scenarios and to study the validity of the SM-EFT framework – here we leave out these complications. For consistency, we compute the thermal corrections to the CPV operator using the same approximations, as discussed in the next section.

At very high temperatures the effective potential only has a minimum at , while for lower temperatures a second minimum appears. In a potential that allows for a first-order EWPT the two minima are degenerate at some critical temperature . The value of the field in the second minimum is denoted by . We find degenerate minima for in the range , in agreement with Refs. [14, 17].

The EWPT proceeds by the formation of bubbles of broken vacuum. If larger than some critical size, these bubbles expand and eventually fill up the entire universe. While bubbles can already form at the critical temperature, their rate may be too small for the phase transtion to complete. The temperature at which tunneling to the true vacuum proceeds is called the nucleation temperature . To obtain this temperature we follow the discussion in Refs. [17, 41].

The tunneling rate is , with the Euclidean action for the so-called bounce solution [42]. At temperatures greater than the inverse bubble radius , the bounce solution is -symmetric [43] and obeys the equation

(29)

with boundary conditions

(30)

gives the Higgs field profile of a static bubble, with the distance from the center of the bubble. The corresponding Euclidean action factorizes into , with

(31)

Nucleation happens when the probability of creating a single bubble within one horizon is of order one [44], which leads to the condition

(32)

The value of the field in the true minimum at is denoted by .

We use the Mathematica Package “AnyBubble” [45] to solve the bounce equation (29) and compute for and . Fig. 2 shows as a function of temperature. For , the minimum of the potential at persists until , which is reflected in the figure by the lower bound on . The nucleation rate is never large enough, and gets trapped in the symmetric vacuum. For the minimum at changes into a maximum before bubbles have had time to nucleate, and the EWPT is not first order.

Figure 2: as a function of temperature for three values of . The horizontal line indicates , the approximate value for which bubbles nucleate. The graph shows that nucleation is impossible for .

In the standard picture of EWBG, a chiral asymmetry is created in front of the bubble wall, which is converted into a baryon asymmetry by sphaleron transitions [46, 47, 48]. In order to preserve the generated baryon asymmetry in the broken phase, the sphaleron transitions should be suppressed inside the bubble. The rate of sphaleron transitions inside the bubble is proportional to , with sphaleron energy being proportional to . We therefore demand the additional condition for baryogenesis and refer to Refs. [49, 50] for a more detailed discussion. We find that this is automatically assured for all values of for which a first-order phase transition is possible in the first place. The strength of the phase transition and the value of increases with .

To summarize, only for a narrow range of values for do we satisfy all criteria for successful baryogenesis:

(33)

If we write this corresponds to the scale .

Finally, we briefly discuss the bubble profile which is needed to calculate the baryon asymmetry. The bounce solution is the initial time () bubble profile. In the rest frame of the bubble, the solution at later times is with , with the radial velocity of the bubble wall. We can define a new variable

(34)

with the location of the bubble wall defined via . In terms of this new coordinate the bubble wall is located at , with the broken phase at and the symmetric phase at , which matches a convention often used in the literature. We can now write the profile solution as a function of . To calculate the baryon asymmetry the wall curvature is usually neglected, and the bubble is approximated by a plane located at ; in this approximation can be replaced by the coordinate perpendicular to the wall, and is extended to . The value of the bounce solution for does not exactly equal , but has a somewhat smaller value. The difference between and is larger when there is a large difference between the potential in the true and the false vacuum.

In the literature the bubble profile is often parametrized by a kink solution [51]

(35)

where is a measure of the width of the bubble wall. The numerical solution can be fit to this parametrization to extract . The kink solution is easy to use, and for scenario A we obtain a baryon asymmetry that only differs from the numerical bounce solution by roughly . In scenario B, however, where the baryon asymmetry depends on the Laplacian of , the kink solution gives very different results. The reason is that the Laplacian contains a term , which, when integrated over , is only convergent because of the boundary conditions in Eq. (30), which guarantee that goes to zero at . The kink solution, however, does not satisfy the boundary condition exactly and consequently the integral diverges. The divergence may be tamed by a suitable regulator444For example, one can add an extra term to the tanh-profile in Eq. (35) that is small in the bubble wall region, but cancels the divergency at the center ., but we will not follow this approach here. To avoid the divergence in scenario B, we will not apply the kink solution for the bubble profile, but instead use the numerical bounce solution in Sect. 5.

The numerical results presented in Sect. 5 are for the benchmark bubble profile, with parameters

(36)

The value for corresponds to a cutoff scale and we have checked that other values of consistent with a first-order EWPT lead to similar conclusions. The value of the numerical bounce solution for is given by GeV. Fitting to the kink solution, we estimate the width of the bubble wall to be . In vacuum the bubble wall would expand at the speed of light, but plasma interactions will reduce the bubble wall velocity. The calculation of is beyond the scope of this paper, we will use the benchmark value given above [52, 53, 54].

4 The matter-antimatter asymmetry

All three Sakharov conditions needed for the creation of a matter-antimatter asymmetry are present in the two scenarios outlined in Sect. 2. The first-order EWPT proceeds via the nucleation of bubbles of the new vacuum, which is an out-of-equilibrium process. The left- and right-handed top quarks in the plasma scatter off the bubble wall differently due to the CPV interactions in Eqs. (7) and (8). As a result, a chiral asymmetry is built in front of the bubble wall. The SM sphaleron transitions only act on the left-handed particles, and transform the chiral asymmetry into a baryon asymmetry. The net baryon charge thus created is swept up by the expanding bubble, and remains conserved provided the phase transition is strong enough such that sphaleron transitions are suppressed in the broken phase inside the bubble.

4.1 Source term

The number densities of the plasma particles in the presence of an expanding bubble are governed by transport equations. The equations for the top quark will include a CPV source term that drives the chiral asymmetry. Here we will just sketch the derivation, focusing on how this source term depends on the bubble wall profile. More details can be found in Ref. [55], whose methods we follow.

The quantum transport equations are derived in the finite temperature Closed-Time-Path formalism [56, 57, 58, 59, 60, 61]. Starting from the Schwinger-Dyson equation a transport equation for the number current of top quarks can be derived

(37)

with for the left- and right-handed top quark respectively. Here are the fermionic Wightman functions (see [55] for the explicit definitions), and the corresponding self-energies defined below in Eq. (41).

It is easiest to work in the rest frame of the bubble, where the Higgs profile is only a function of as given in Eq. (34), and we can express all space-time derivatives in terms of -derivatives. In the diffusion approximation the current can be written as with the number densities and the diffusion coefficient (see Eq. (B.2)). In addition, we neglect the curvature of the bubble wall, and model the bubble wall as a plane located at . With these approximations

(38)

where the last expression is valid for the planar approximation, and where a prime denotes a derivative with respect to .

In the bubble background the top quark mass is space-time dependent as it depends on the Higgs background . To deal with this complication, the self-energies are calculated in the “vev-insertion approximation” [62, 63, 64, 65], which amounts to treating the field dependent part of the top mass as a perturbation. To compare the asymmeties produced in scenarios A and B in a consistent way it is important to work at the same order in perturbation theory in both the Higgs and the CPV sector. Thus we include the one-loop thermal corrections to the CPV interactions, which are calculated in Appendix A, and neglect daisy diagrams. The zero-temperature top mass555In Sect. 2.1, we used the symbol to denote the real top mass at zero temperature, which is relevant for the EDM calculation. In the current section, however, is a complex number. can be split into a real and imaginary part (indicated by superscripts), and likewise for the thermal corrections . The quadratic Lagrangian for the top quarks is split into a free part, independent of the bubble profile, and a field-dependent interaction part, according to

(39)
(40)

The -functions defined above, which parameterize the interaction strength, are derived in Appendix A for the scenarios under investigation. are the usual SM thermal masses [66], which we list in Eq. (B.3). They can be viewed as one-loop thermal corrections to the massless propagator. Since these corrections do not depend on the space-time dependent Higgs profile, they can be resummed and included in the full propagator , which is constructed from the free Lagrangian. are the one-loop thermal corrections to the CPV -vertex. All the terms in are field dependent, and therefore treated as a perturbation. The imaginary part of the top mass is space-time dependent in the bubble background, and cannot be rotated away by a chiral transformation if it is non-linear in the field. Its presence leads to different dispersion relations for left- and right-handed particles, and consequently different forces act on them as they scatter with the bubble wall. This is the physical underpinning of the appearance of a source term, denoted by , in the transport equations that drives the chiral asymmetry. Based on this discussion, we expect , as it should be proportional to , depend on the phase of , and be quadratic in as the diagram for scattering requires at least two mass insertions. This is confirmed by the explicit derivation, which we will now sketch.

We consider the transport equation for the right-handed top quark . The self-energy obtains a contribution from the diagram with two mass insertions

(41)

with the left- and right-handed projection operators. Using Eq. (41) in the transport equation, Eq. (37), we can separate the right-hand side into a real and imaginary part, corresponding to the CP-conserving relaxation term and the CPV source

(42)

where we used the short-hand . The subscript indicates that mass can be set to zero in the trace of the propagators666Inserting Eq. (41) in Eq. (37) gives a trace of a product of propagators and projection operators, which in Fourier space is of the form . By defining (43) Eq. (42) can be neatly split into a CP-conserving and CPV part.. The analagous equation can be written down for the left-handed quark, with and .

In the limit that the typical time scale for thermalization of the top quarks is much faster than the time scale on which the Higgs profile changes, we can expand777Here we used that Taylor expanding , the term vanishes when substituted in the integral in Eq. (42) because of spatial isotropy, and thus only the term proportional to the time-derivative contributes [55].

(44)

and the -dependent parts can be taken outside the -integral in Eq. (42). This gives the result we are after, as it factors out the explicit dependence on the bubble-wall profile. We thus find that and , with the constant of proportionality a function of the temperature, thermal masses , and top decay width only, as these are the quantities entering the propagator. Moreover, the thermal corrections to the CPV operator, and thus to the source, and the effective potential are calculated consistently.

4.2 Transport equations

To calculate the chiral asymmetry in front of the bubble wall we keep track of the number density of the third-generation quarks and the Higgs field. The electroweak gauge interactions are fast, and approximate chemical equilibrium between the members of the left-handed doublet is assumed. Consider then the following densities , , and , with the number density of quarks minus anti-quarks, and for the real Higgs field the number density of Higgs particles. Since the CP violation resides purely in the top quark sector888We neglect the CPV top-electron coupling that appears in scenario B (see Eq. (8)) as it is proportional to the small electron Yukawa coupling., no asymmetry is built up in the lepton sector. The first- and second-generation quarks only interact via strong sphaleron processes on the relevant time scales, and their densities can be related to those of the third generation. The total chiral asymmetry is [67]. Because of the different time scales involved we can describe the creation of the chiral asymmetry, and the transformation into a net baryon asymmetry as a two-step process.

The set of coupled transport equations can be derived as explained in the previous section. In addition to the relaxation and source term from the mass-insertion diagrams, there are Yukawa interactions that contribute to . The (non-perturbative) strong sphaleron interactions are also included. The full set of transport equations is [55]

(45)

All rates and input parameters needed to solve this set of equations are given in Appendix B. , and are the strong sphaleron rate, the Yukawa interaction rate, and the relaxation rate, respectively. The latter two are extracted from and, as discussed in the previous subsection, are proportional to . The difference between scenario A and B lies thus solely in the source term, which we give here explicitly999In the expressions for , and we have neglected the collective plasma hole excitations to the propagators [68, 69, 70].

(46)

where denotes the Fermi-Dirac distribution, , and the top decay width. The “-1” term in the numerator on the second line gives a divergent contribution that survives in the zero-temperature limit where the distributions are Boltzmann suppressed. This divergence is absorbed by the counterterms of the zero-temperature renormalized action, or equivalently, this term can be removed by normal ordering the operators [71].

Assuming local thermal equilibrium and small chemical potentials, the -functions are implicitly defined via (see Eq. (B.4) for more details)

(47)

where the mass can be approximated by the real part 101010Since the r.h.s. of the transport equation is calculated using the vev-insertion approximation, it can be argued that the mass used in the -functions should be the thermal mass instead, i.e. . Doing so would only give a small difference in the final asymmetry.. Furthermore we have

(48)

which is often approximated by [67].

The set of transport equations Eq. (45) reduces to ordinary differential equations in the approximation of Eq. (38), and can be solved to find the net chiral assymmetry . The SM sphalerons convert this into a net baryon number. Integrating over the asymmetric phase , where the sphalerons act, the baryon asymmetry becomes