# Resummation and cancellation of the VIA source in electroweak baryogenesis

Marieke Postma b Jorinde van de Vis c and Graham White<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Nikhef, Science Park 105,
1098 XG Amsterdam, The NetherlandsDeutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, GermanyKavli IPMU (WPI), UTIAS, The University of
Tokyo, Kashiwa, Chiba 277-8583, Japan

###### Abstract

We re-derive the vev-insertion approximation (VIA) source in electroweak
baryogenesis. In contrast to the original derivation, we rely solely on
1-particle-irreducible self-energy diagrams. We solve the Green’s function
equations both perturbatively and resummed over all vev-insertions. The VIA
source corresponds to the leading order contribution in the gradient expansion
of the Kadanoff-Baym (KB) equations. We find that it vanishes both for bosons
and fermions, both in the perturbative and in the resummed approach.
Interestingly, the non-existence of the source is a result of a cancellation
between different terms in the KB equations, and not of a pathology in the
vev-insertion approximation itself.

††preprint: Nikhef 2022-007, DESY-22-092, IPMU22-0034

##  1 Introduction

In electroweak baryogenesis (EWB) the observed matter-antimatter asymmetry in
the Universe is created during a first order electroweak phase transition
Kuzmin:1985mm ; Shaposhnikov:1987tw ; McLerran:1990zh ; Cohen:1990it  (for a
review, see Cohen:1993nk ; Trodden:1998ym ; Cline:2006ts ; Morrissey:2012db ;
White:2016nbo ; Garbrecht:2018mrp ). In order to satisfy the Sakharov
conditions (violation of baryon number, charge (C) and charge-parity (CP)
symmetry, and out-of-equilibrium dynamics), new physics at the electroweak
scale is required Sakharov:1967dj . Due to the relatively low energy scale,
these new interactions can be probed by experiments. For example, the latest
ACME results ACME:2018yjb  on electric dipole moments put stringent
constraints on CP violation in effective Yukawa interactions between the Higgs
field and the Standard Model (SM) fermions, already ruling out a large class
of models (see e.g. Cline:2011mm ; Liebler:2015ddv ; Dorsch:2016nrg ;
Cline:2017jvp ; deVries:2017ncy ; Bodeker:2020ghk ).

Unfortunately, the theoretical uncertainties in the computation of the baryon
asymmetry are large, as different approaches lead to predictions that may vary
by orders of magnitude Cline:2017jvp ; Cline:2020jre ; Cline:2021dkf . The
main difference between methods is in the derivation of the source terms in
the transport equations. In particular, using the vev-insertion approximation
(VIA), models with CP-violating Yukawa couplings may still be marginally
consistent with the electric dipole bounds DeVries:2018aul ; Fuchs:2020uoc ,
whereas in a semi-classical computation the corresponding asymmetry is orders
of magnitude too small Cline:2021dkf . In this paper we will take a careful
look at the derivation of the VIA source term in the transport equations.

The transport equations describe the evolution of the phase space densities of
the plasma particles as they interact with the expanding bubbles during the
electroweak phase transition. The Higgs vacuum expectation value (vev) varies
from zero in the false vacuum outside the bubble to some finite value
inside.111In multi-field or multi-step transitions some combination of the
scalar fields in the extended Higgs sector has a changing vev inside the
bubble wall. This leads to a spacetime-dependent mass for the SM fermions and
other degrees of freedom with couplings to the Higgs field, and consequently
to a semi-classical force Joyce:1994zt ; Cline:2000nw ; Kainulainen:2001cn ;
Prokopec:2003pj ; Prokopec:2004ic . In a multi-flavor set-up the mixing matrix
between the flavor and mass eigenstates varies inside the bubble wall as well
Konstandin:2004gy ; Konstandin:2005cd ; Cirigliano:2011di . In the presence of
CP violation, both effects can give a chiral asymmetry in front of the bubble
wall, which is generated by appropriate source terms in the transport
equations. Electroweak sphaleron transitions then transform the chiral
asymmetry into a baryon asymmetry.

The vev-insertion approximation treats the space-time-dependent mass term as
an interaction term, and calculates the source term in a perturbative
expansion Riotto:1995hh ; Riotto:1997vy . Unlike the semi-classical and flavor
source terms, there is no clear physical explanation for its origin. The VIA
source crucially depends on thermal corrections, in particular on the thermal
width, making it distinct from the other sources. Flavor dynamics is essential
as well, as for example in a set-up with a left- and right-handed fermion, the
left-handed source term is minus the right-handed source. Note that the use of
the term ‘flavor’ is somewhat cavalier in this context, and used to
distinguish different chiralities. We nevertheless stick to this terminology
to highlight the similarities between a bosonic and fermionic source term; in
both cases the source hinges on a misalignment between the propagating state
and the state that interacts with the plasma. The VIA source already arises at
leading order in the derivative expansion, which treats the bubble wall
background as evolving slowly compared to all other relevant time scales. This
explains why it generates larger asymmetries compared to other theoretical
approaches Cline:2020jre ; Cline:2021dkf .

The authors of Ref. Carena:2000id  tried to avoid the VIA expansion, but their
approach is not based on a first principle derivation, but rather on a
phenomenological method, including a phenomenological decay width, following
Huet:1995sh ; Riotto:1998zb . In Ref. Cline:2020jre  it was noted that for
degenerate masses the source term diverges in the zero width limit. However,
the approximations made to derive the VIA source already break down before
this limit is reached Postma:2021zux . Ref. Kainulainen:2021oqs  highlighted
technical issues with the VIA source, such as omitted pole contributions and
renormalization issues. While these indeed should be properly addressed, they
do not directly invalidate the derivation, as the renormalization issue does
not arise in a bosonic system, and the extra pole contributions _can_ be
included. More fundamentally compromising for the VIA approach, Ref.
Kainulainen:2021oqs  also showed that VIA-like sources vanish at all orders in
the derivative expansion if the interactions with the thermal bath are
mediated by QCD. This result can be understood because the VIA source arises
from the axial current equation, which vanishes in a vector-like theory.
Indeed, all equations can be written in terms of the phase space densities of
Dirac fermions, and thus no VIA source that is equal but opposite in sign for
the left- and right-handed fermions is generated. In this paper we allow for
chiral interactions with the bath which can yield a non-zero axial current, in
line with the implicit assumption usually made in the VIA literature.

Although the VIA-method already appeared in Huet:1994jb ; Huet:1995mm ;
Huet:1995sh , the source term was first derived in the closed-time-path
formalism in Riotto:1995hh ; Riotto:1998zb , and subsequently generalized to
include a CP-conserving relaxation rate in Lee:2004we . Both the CP-violating
and -conserving source terms are derived from the Schwinger-Dyson equations by
expanding the self-energies to second order in the VIA expansion. This results
in a self-energy diagram that is non-local Kainulainen:2021oqs  and not one-
particle-irreducible (1PI) Postma:2021zux . This approach makes it hard to
count diagrams correctly, which is an obstacle to resumming all self-energy
insertions. The source term is derived from the collision term in the
transport equations, which is evaluated with a second order self-energy and
zeroth order Green’s function. In Postma:2021zux  it was demonstrated that the
same source term can be obtained from the mass commutator in the transport
equations, if the self-energy is evaluated at first order – this is a 1PI
diagram – and the Green’s function is computed to first order in VIA as well.
This equivalence strongly suggest the technical issues noted in
Kainulainen:2021oqs  do not arise from the non-local nature of the self-energy
diagrams appearing in the standard derivation.

In this paper, we will follow the approach of Ref. Postma:2021zux  and only
include 1PI, equilibrium, self-energy corrections, which is equivalent to
treating the space-time dependent part of the mass matrix as a perturbation.
The Green’s functions at any given order in VIA then follow from the
Schwinger-Dyson equations (or in Wigner space, from the Kadanoff-Baym
equations). The Green’s functions can be determined perturbatively, but
remarkably, the full set of constraint equations can be solved exactly as
well. The VIA source term is then derived both perturbatively as well as fully
resummed over all mass insertions. We find that the source term actually gets
two contributions, from the mass-commutator and from the collision term in the
transport equations. The cancellation happens order by order in VIA, and also
for the fully resummed case. There is thus no VIA source term at leading order
in the derivative expansion. This cancellation was missed before as the
original non-1PI approach obscures this Riotto:1995hh ; Riotto:1998zb ;
Lee:2004we , and Postma:2021zux  only considered the mass-commutator but not
the collision term.

When something cancels exactly, there usually is a deeper reason. Could we
actually have foreseen our results? We think we could have. In VIA the Green’s
functions are derived at leading order in the derivative expansion, and are
thus indistinguishable from the Green’s functions in a constant background.
The source term is also derived at leading order in the derivative
expansion.222To be precise, at leading order in the expansion of the diamond
operator in the Kadanoff-Baym equations. In the standard VIA approach, in
which this source does not cancel, the integrand is then subsequently expanded
in something similar to a derivative expansion to extract the dominant CP-
violating part – but this is not equivalent to the diamond expansion. In the
limit of a constant background the source should vanish, as equilibrium is
restored. But this is the same limit as taking the zeroth order in the
derivative expansion, and one should thus have expected a zero source.

This paper is organized as follows. Section 2 introduces the Kadanoff-Baym
equations and the Green’s functions that describe the out-of-equilibrium
plasma. In Sections 3 and 4 we solve the Green’s functions with and without
VIA expansion for bosons and fermions respectively. We also derive the
corresponding source term and demonstrate that they are zero. We conclude in
Section 5.

##  2 CTP formalism and Green’s functions

The closed-time path (CTP), or in-in formalism, provides a first-principle
description in terms of Green’s functions333We will use the terms ‘Green’s
function’ and ‘propagator’ interchangeably. of a system out-of-equilibrium.
The Schwinger-Dyson equations can be split into hermitian and anti-hermitian
parts, usually referred to as the constraint and kinetic equations. The
constraint equations provide the spectral information of the system, whereas
the kinetic equation determines the dynamical evolution. In this section we
list the results and relations relevant for the discussion of the source terms
for electroweak baryogenesis. This section also serves to set the notation.
More details and derivations can be found in e.g. Riotto:1998zb ;
Prokopec:2003pj ; Cirigliano:2011di ; Cirigliano:2009yt . We use the same
conventions as Prokopec:2003pj .

The bosonic and fermionic CTP Green’s functions are, respectively,

$i\Delta(u,v)=\langle\Omega|T_{\cal
C}\,\left[\phi(u)\phi^{\dagger}(v)\right]|\Omega\rangle,\qquad
iS_{\alpha\beta}(u,v)=\langle\Omega|T_{\cal
C}\,\left[\psi_{\alpha}(u)\bar{\psi}_{\beta}(v)\right]|\Omega\rangle,$ (1)

with $T_{\cal C}$ denoting time-ordering along the contour ${\cal C}$ which
runs from initial time $t_{0}=-\infty$ to time $t$ and back. The contour can
thus be split into a forward- and backward-going branch, and fields are
labeled by $a=\pm$, depending on the branch they lie on. We define
$G^{ab}=\\{\Delta^{ab},S^{ab}\\}$ for bosons and fermions respectively, and
introduce the notation
$G^{t}=G^{++},\,G^{\bar{t}}=G^{--},\,G^{>}=G^{-+},\,G^{<}=G^{+-}$.

The retarded and advanced propagators are

$\displaystyle G^{r}$ $\displaystyle\equiv G^{t}-G^{<}=G^{>}-G^{\bar{t}},$
$\displaystyle G^{a}$ $\displaystyle\equiv G^{t}-G^{>}=G^{<}-G^{\bar{t}},$ (2)

from which it also follows that $G^{t}+G^{\bar{t}}=G^{>}+G^{<}$. Throughout we
will label $G^{\lambda}$ with $\lambda=>,\,<$ for the so-called Wightman
functions, and $G^{\alpha}$ with $\alpha=r,\,a$ for the retarded and advanced
propagators. The hermiticity properties of the Wightman functions are
$(i\Delta^{\lambda}(u,v))^{\dagger}=i\Delta^{\lambda}(v,u)$ and
$(\gamma^{0}iS^{\lambda}(u,v))^{\dagger}=\gamma^{0}iS^{\lambda}(v,u)$ for
bosons and fermions respectively. The retarded/advanced propagators can be
split into hermitian and anti-hermitian parts via

$G^{r,a}=G^{h}\mp iG^{\cal A}\;\;\;{\rm
with}\;\;\;G^{h}\equiv\frac{1}{2}(G^{r}+G^{a})\;\;\;\&\;\;\;G^{\cal
A}\equiv\frac{1}{2i}(G^{a}-G^{r}),$ (3)

with the upper (lower) sign for the retarded (advanced) propagator in the
first expression. In thermal equilibrium, the Wightman functions can be
written in terms of the spectral function444The fermionic Wightman functions
can be projected onto a helicity basis, and eq. 4 then applies to the
projected Wightman functions, see eqs. 75 and 76 for the exact expressions for
fermions.

$G^{\lambda}=\rho g^{\lambda},\quad{\rm with}\quad
g^{>}=(1+sn_{s})\;\;\;\&\;\;\;g^{<}=sn_{s}\quad{\rm
and}\quad\rho\equiv(G^{r}-G^{a}),$ (4)

with $\rho$ the spectral function. Here $s=1$ for bosons and $s=-1$ for
fermions, and $n_{s}=({\rm e}^{(E-\mu)/T}-s)^{-1}$ is the Bose-Einstein or
Fermi-Dirac distribution for bosons and fermions, respectively.

The temperature-dependent self-energies arise from loop corrections in the
plasma background. To emphasize the similarity between the bosonic and
fermionic system, we will denote the self-energies for both by $\Pi^{ab}$,
rather than using the more conventional $\Sigma^{ab}$ notation for the
fermionic self-energy. When approximated by their thermal equilibrium
expressions, the self-energies $\Pi^{ab}$ satisfy the same relations as the
Green functions $G^{ab}$ listed above. In particular, the advanced/retarded
self-energy can be split into real and imaginary parts $\Pi^{r,a}=\Pi^{h}\mp
i\Pi^{\cal A}$, with $\Pi^{h}$ containing the thermal mass corrections and
$\Pi^{\cal A}$ the thermal widths.

###  2.1 Kadanoff-Baym equations

The position-space Schwinger-Dyson equations can be written in the form
Riotto:1998zb ; Prokopec:2003pj :

$\displaystyle K(u)G^{ab}(u,v)$
$\displaystyle=a\delta_{ab}\delta^{4}(u-v)+\sum_{c}c\int
d^{4}w\,\Pi^{ac}(u,w)G^{cb}(w,v),$ (5)

with $a,b,c=\pm$. For bosons $K(u)=-(\partial^{2}_{u}+M_{0}(u)^{2})$ is the
Klein-Gordon operator with $M_{0}^{2}$ the tree-level mass matrix, and for
fermions $K(u)=(i\not{\partial}_{u}-M_{0}(u))$ is the Dirac operator with
$M_{0}(u)$ the tree-level Dirac mass.

For thick bubble walls, one can perform a gradient expansion in the slowly
varying bubble background. It is then useful to define the relative and
center-of-mass coordinates $r=u-v$ and $x=\frac{1}{2}(u+v)$, and to make a
Wigner transform, which is defined as the Fourier transform with respect to
the relative coordinate:

$G(k,x)=\int{\rm d}^{4}(u-v)\,{\rm e}^{ik.(u-v)}G(u,v)=\int{\rm d}^{4}r\,{\rm
e}^{ik.r}G(x+\frac{1}{2}r,x-\frac{1}{2}r).$ (6)

The gradient expansion in the slowly varying background is then equivalent to
an expansion in the diamond operator, which is defined as

$\diamond\big{(}A(k,x)B(k,x)\big{)}=\frac{1}{2}\big{(}\partial_{x}A(k,x)\cdot\partial_{k}B(k,x)-\partial_{k}A(k,x)\cdot\partial_{x}B(k,x)\big{)}.$
(7)

Wigner transforming the hermitian and anti-hermitian part of the Schwinger-
Dyson equation (5) gives the constraint and kinetic equations, known as the
Kadanoff-Baym (KB) equations. For bosons this gives

$\displaystyle(k^{2}-\frac{1}{4}\partial_{x}^{2})G^{ab}$
$\displaystyle=a\delta_{ab}\otimes 1+\frac{1}{2}{\rm
e}^{-i\diamond}\Big{(}\\{M_{0}^{2},G^{ab}\\}+\sum_{c}c(\Pi^{ac}G^{cb}+G^{ac}\Pi^{cb})\Big{)},$
$\displaystyle 2ik\cdot\partial_{x}G^{ab}$ $\displaystyle={\rm
e}^{-i\diamond}\Big{(}[M_{0}^{2},G^{ab}]+\sum_{c}c(\Pi^{ac}G^{cb}-G^{ac}\Pi^{cb})\Big{)}.$
(8)

Note that $G^{ab},\Pi^{ab},M_{0}^{2}$ and $1$ are $n\times n$-matrices in
flavor space with $n$ complex bosonic degrees of freedom. For fermions we get

$\displaystyle\frac{1}{2}\\{\not{k},G^{ab}\\}$
$\displaystyle=a\delta_{ab}\otimes 1+\frac{1}{2}{\rm
e}^{-i\diamond}\Big{(}\\{M_{0},G^{ab}\\}+\sum_{c}c(\Pi^{ac}G^{cb}+G^{ac}\Pi^{cb})\Big{)},$
$\displaystyle\frac{i}{2}\\{\not{\partial}_{x},G^{ab}\\}$ $\displaystyle={\rm
e}^{-i\diamond}\Big{(}[M_{0},G^{ab}]+\sum_{c}c(\Pi^{ac}G^{cb}-G^{ac}\Pi^{cb})\Big{)}.$
(9)

For a single Dirac fermion $G^{ab},\Pi^{ab},\delta M$ and $1$ are matrices in
Dirac space; projecting onto left and right chiralities, this can be viewed as
$2\times 2$ in ‘flavor’ space.

###  2.2 Retarded/Advanced propagator and thermal corrections

Introducing the matrices

$\tilde{G}=\left(\begin{array}[]{cc}G^{t}&-G^{<}\\\
G^{>}&-G^{\bar{t}}\end{array}\right),\qquad\tilde{\Pi}=\left(\begin{array}[]{cc}\Pi^{t}&-\Pi^{<}\\\
\Pi^{>}&-\Pi^{\bar{t}}\end{array}\right),$ (10)

the Schwinger-Dyson eq. 5 takes the simple form Riotto:1998zb

$\displaystyle K(u)\tilde{G}(u,v)$ $\displaystyle=\delta^{4}(u-v)+\int
d^{4}w\,\tilde{\Pi}(u,w)\tilde{G}(w,v).$ (11)

For a single flavor system in thermal equilibrium the self-energies and
Wightman functions satisfy the KMS relations

$(1+sn_{s})G^{<}=sn_{s}G^{>},\quad(1+sn_{s})\Pi^{<}=sn_{s}\Pi^{>}.$ (12)

We note that in the presence of flavor mixing the equilibrium conditions are
no longer of the simple KMS form eq. 12. We will return to this point in
section 3.4.1.

We can diagonalize the propagator and self-energy matrices by an orthogonal
matrix $V$ (with $VV^{-1}=1$)

$V\tilde{G}V^{-1}=\left(\begin{array}[]{cc}G^{r}&0\\\
0&G^{a}\end{array}\right),\quad
V\tilde{\Pi}V^{-1}=\left(\begin{array}[]{cc}\Pi^{r}&0\\\
0&\Pi^{a}\end{array}\right),\quad{\rm
with}\;\;V=\left(\begin{array}[]{cc}(1+sn_{s})&-sn_{s}\\\
1&-1\end{array}\right).$ (13)

The diagonal entries $G^{r,a}$ and $\Pi^{r,a}$ are the retarded/advanced
propagators and self-energies respectively. Eqs. (10) and (13) contain the
same information as the relation between the Green’s functions in eqs. 2 and
4. Using this rotation, the Schwinger-Dyson equations, and thus also the
Wigner space KB equations, can be diagonalized.

##  3 Bosons

In this section we will derive the VIA source for a set-up with two flavors
and an off-diagonal spacetime-dependent mass. We assume that the interactions
with the bath are flavor-diagonal and are different for the two flavors. The
results can be generalized to other bosonic systems as well. After discussing
the set-up and the thermal corrections, we will first solve the constraint
equation and derive the VIA source perturbatively, expanding in the number of
mass insertions. However, the constraint equations can also be solved exactly,
without such an expansion, and we also present the corresponding resummed
results.

###  3.1 Lagrangian

For definiteness, consider a toy model with two bosonic flavors, labeled by
$L,R$, with a CP-violating interaction with the Higgs field. In terms of the
flavor doublet $\phi=1/\sqrt{2}(\phi_{L}\;\phi_{R})^{T}$, the quadratic
Lagrangian in the bubble wall background is

$\mathcal{L}^{(2)}=(\partial_{\mu}\phi)^{\dagger}(\partial^{\mu}\phi)-\phi^{\dagger}M_{0}^{2}\phi,\qquad
M_{0}^{2}(v)=\left(\begin{array}[]{cc}m_{LL,0}^{2}&m_{LR}^{2}(v)\\\
m_{RL}^{2}(v)&m_{RR,0}^{2}\end{array}\right).$ (14)

The subscript $0$ indicates that this is the zero temperature mass matrix. The
field-independent diagonal mass terms will receive finite temperature
corrections, which are diagonal in flavor space for a flavor-diagonal coupling
to the heat bath. The off-diagonal masses depend on the background Higgs vev
$v$, which is spacetime dependent, and violates CP in the bubble wall
background. The vev insertion approximation consists of treating
$m^{2}_{LR}=(m^{2}_{RL})^{*}$ as a perturbation, and expanding the KB-
equations in this small quantity.

###  3.2 KB equations

The constraint equations for the Wightman functions $G^{\lambda}$ with
$\lambda=>,\,<$ and time-ordered propagator $G^{t}$ follow from eq. 8

$\displaystyle\left(k^{2}-\frac{1}{4}\partial_{x}^{2}\right)G^{\lambda}$
$\displaystyle=\frac{1}{2}{\rm
e}^{-i\diamond}\left(\\{M^{2},G^{\lambda}\\}+\\{\Pi^{\lambda},G^{h}\\}+\frac{1}{2}([\Pi^{>},G^{<}]-[\Pi^{<},G^{>}])\right),$
$\displaystyle\left(k^{2}-\frac{1}{4}\partial_{x}^{2}\right)G^{t}$
$\displaystyle=1+\frac{1}{2}{\rm
e}^{-i\diamond}\left(\\{M^{2}+\Pi^{t}-\Pi^{h},G^{t}\\}-\Pi^{<}G^{>}-G^{<}\Pi^{>}\right).$
(15)

Here we rewrote the right-hand side of the first equation using the following
relations (and the equivalent ones with $\Pi$ and $G$ flipped)

$\displaystyle\Pi^{>}G^{t}-\Pi^{\bar{t}}G^{>}$
$\displaystyle=\Pi^{h}G^{>}+\Pi^{>}G^{h}+\frac{1}{2}(\Pi^{>}G^{<}-\Pi^{<}G^{>}),$
$\displaystyle\Pi^{t}G^{<}-\Pi^{<}G^{\bar{t}}$
$\displaystyle=\Pi^{h}G^{<}+\Pi^{<}G^{h}+\frac{1}{2}(\Pi^{>}G^{<}-\Pi^{<}G^{>}).$
(16)

We have further defined $M^{2}=M_{0}^{2}+\Pi^{h}$, where $M_{0}^{2}$ is the
tree-level mass matrix, and $\Pi^{h}$ the thermal mass, as shown in the next
subsection. Note that since the system of equations only depends on $M^{2}$,
it is fully equivalent to include the space-time dependent mass in $M_{0}^{2}$
and expand the mass matrix, or to include it in the interaction Lagrangian and
add the 1PI diagram with one mass insertion to $\Pi^{h}$ Postma:2021zux . Here
we follow the former approach.

The kinetic equation for the Wightman functions is

$\displaystyle 2ik\cdot\partial_{x}G^{\lambda}$ $\displaystyle={\rm
e}^{-i\diamond}\Big{(}[M^{2},G^{\lambda}]+[\Pi^{\lambda},G^{h}]+\frac{1}{2}\left(\\{\Pi^{>},G^{<}\\}-\\{\Pi^{<},G^{>}\\}\right)\Big{)}.$
(17)

Integrating the left-hand side over four-momenta and adding up the two
different Wightman functions gives Riotto:1998zb

$\frac{1}{2}\partial_{\mu}\int\frac{{\rm
d}^{4}k}{(2\pi)^{4}}ik^{\mu}\left(G^{>}(k,x)+G^{<}(k,x)\right)=-i\langle\phi^{\dagger}(x)\overset{\leftrightarrow}{\partial^{\mu}}\phi(x)\rangle=-\partial_{\mu}\langle
J^{\mu}(x)\rangle.$ (18)

We can then identify the source term for the left- and right-handed flavors
from the right-hand side of the kinetic equation. The integration over
4-momenta only picks out the leading order term in the diamond expansion,
which amounts to setting ${\rm e}^{-i\diamond}=1$. Explicitly, the left-handed
source is

$\displaystyle S_{LL}$ $\displaystyle=-\int\frac{{\rm
d}^{4}k}{(2\pi)^{4}}\left([M^{2},G^{>}+G^{<}]+[\Pi^{>}+\Pi^{<},G^{h}]+\\{\Pi^{>},G^{<}\\}-\\{\Pi^{<},G^{>}\\}\right)_{LL}$
$\displaystyle\equiv-\int\frac{{\rm d}^{4}k}{(2\pi)^{4}}\bar{S}_{LL},$ (19)

where the label $LL$ refers to the $(1,1)$-component of the resulting source
matrix. We introduced the notation $\bar{S}_{LL}$ for the integrand for future
convenience.

###  3.3 Thermal corrections

Let’s first consider the solutions of the constraint equation for a flavor-
diagonal mass matrix and self-energy, and see how the thermal self-energies
affect the solution. This would be the starting point in the vev-insertion
approximation, where at leading order the off-diagonal mass terms are set to
zero. The constraint equations for different flavors decouple, and we can
focus on a single flavor. We work at leading order in the derivative expansion
and set ${\rm e}^{-i\diamond}=1$.

If the thermal plasma is close to equilibrium, to first approximation the
thermal self-energies and propagators satisfy the KMS relations eq. 12. We can
then rotate the KB equations to the basis of retarded and advanced propagators
using eq. 13:

$k^{2}G^{\alpha}-\frac{1}{2}\\{M_{0}^{2}+\Pi^{\alpha},G^{\alpha}\\}=(k^{2}-M_{0}^{2}-\Pi^{\alpha})G^{\alpha}=1,$
(20)

with $\alpha=r,a$. The self-energies for each flavor can be rewritten in terms
of the hermitian and anti-hermitian parts eq. 3, which are related to the
thermal mass $m_{T}$ and width $\Gamma_{T}$ respectively, via Greiner:1998vd ;
Riotto:1998zb ; Henning:1996dv

$\displaystyle\Pi^{\alpha}=\Pi^{h}\mp i\Pi^{\cal A},\quad{\rm
with}\quad\Pi^{h}=m^{2}_{T}\;\;\;\&\;\;\;\Pi^{\cal A}=2k_{0}\Gamma,$ (21)

with the upper (lower) sign for the retarded (advanced) propagator. The sign
(and factor 2) in the bosonic case are taken to obtain the same result as in
Riotto:1995hh ; Lee:2004we  for the spectral function. The constraint equation
for a single flavor is solved by

$\displaystyle(G^{\alpha})^{-1}$ $\displaystyle=k^{2}-m_{0}^{2}-\Pi^{h}_{T}\pm
i\Pi_{T}^{\cal A}.$ (22)

The Wightman functions are given by eq. 4 in terms of the spectral function

$\rho=G^{r}-G^{a}=\frac{-2i\Pi^{\cal
A}}{(k^{2}-m_{0}^{2}-\Pi^{h})^{2}+(\Pi^{\cal
A})^{2}}=\frac{\gamma}{(k^{2}-m^{2})^{2}-\gamma^{2}/4},$ (23)

with $m^{2}=m_{0}^{2}+\Pi^{h}$ the tree-level plus thermal mass, and
$\gamma=-4ik_{0}\Gamma$.

###  3.4 Vev-insertion approximation

Just as in the standard VIA approach, we work at leading order in the
derivative expansion and set ${\rm e}^{-i\diamond}=1$ in the KB equations. In
VIA the off-diagonal mass term is treated as a perturbation, and one can solve
the KB equations perturbatively as a series in the number of ‘mass
insertions’, i.e. vev insertions Riotto:1995hh ; Riotto:1997vy ; Riotto:1998zb
; Lee:2004we . The solution of the constraint equations can be substituted
into the kinetic equation to get the dynamics of the system and derive the
source term eq. 19. The source term is calculated at 2nd order in VIA, which
requires solving the constraint equation to 2nd order as well.

####  3.4.1 Constraint equation

The mass matrix can be expanded as $M^{2}=M_{d}^{2}+\delta M^{2}$, with
$M_{d}$ containing the diagonal tree-level plus thermal masses, and $\delta
M^{2}$ the off-diagonal and field-dependent masses. The constraint equations
at leading order in VIA – i.e. setting $\delta M^{2}=0$ – are

$\displaystyle k^{2}G_{(0)}^{\lambda}$
$\displaystyle=\frac{1}{2}\Big{(}\\{M_{d}^{2},G_{(0)}^{\lambda}\\}+\\{\Pi^{\lambda},G_{(0)}^{h}\\}+\frac{1}{2}([\Pi^{>},G_{(0)}^{<}]-[\Pi^{<},G_{(0)}^{>}])\Big{)},$
$\displaystyle k^{2}G_{(0)}^{t}$
$\displaystyle=1+\frac{1}{2}\Big{(}\\{M_{d}^{2}+\Pi^{t}-\Pi^{h},G_{(0)}^{t}\\}-\Pi^{<}G_{(0)}^{>}-G_{(0)}^{<}\Pi^{>}\Big{)},$
(24)

where we have set ${\rm e}^{-i\diamond}=1$. We have also dropped the
$\frac{1}{4}\partial_{x}^{2}G_{(0)}^{\lambda}$ term, as this term likewise
leads to derivatives of the background dependent mass, which is assumed to be
sub-leading in the derivative expansion. This is not different from the
standard assumption in the VIA computation. The thermal corrections arise from
(approximately) thermal equilibrium physics, and we assume that the self-
energies satisfy KMS relations, and are flavor diagonal:

$\Pi_{IJ}^{\lambda}=g^{\lambda}_{II}\gamma_{I}\delta_{IJ},\qquad\gamma_{I}=-4ik_{0}\Gamma_{I}.$
(25)

with $I,J=L,R$ and $g^{\lambda}$ given in eq. 4. The off-diagonal self-
energies vanish.

The thermal self-energies are given in eq. 25. The 0th order KB equations are
solved by Wightman functions that are flavor diagonal and that satisfy the KMS
relations

$G_{(0),IJ}^{\lambda}=g^{\lambda}_{II}\rho_{(0),I}\delta_{IJ}.$ (26)

Substituting this and eq. 25 into the leading order constraint equations (24),
one can solve for the spectral function555At the pole
$k_{0}\approx\omega_{I}=\sqrt{|\mathbf{k}|^{2}+m_{I}^{2}}$ and we can write
the spectral function in the form  $\displaystyle\rho_{(0),I}$
$\displaystyle=\frac{1}{2\omega_{I}}\left[\left(\frac{1}{k^{0}-\omega_{I}+i\Gamma_{I}}-\frac{1}{k^{0}+\omega_{I}+i\Gamma_{I}}\right)-\left(\frac{1}{k^{0}-\omega_{I}-i\Gamma_{I}}-\frac{1}{k^{0}+\omega_{I}-i\Gamma_{I}}\right)\right].$
This is the standard form of the zeroth order propagator used in the VIA
approach Lee:2004we .

$\displaystyle\rho_{(0),I}$
$\displaystyle=\frac{\gamma_{I}}{(k^{2}-m_{I}^{2})^{2}-\gamma_{I}^{2}/4}=\frac{\gamma_{I}}{D_{I}D^{*}_{I}},$
(27)

with $D_{I}=k^{2}-m_{I}^{2}-\gamma_{I}/2$ and
$D^{*}_{I}=k^{2}-m_{I}^{2}+\gamma_{I}/2$. The spectral function agrees with
the result derived in the basis of retarded and advanced propagators in
thermal equilibrium eq. 23. The solution for the time-ordered Green’s function
is

$G_{(0),IJ}^{t}=\left[n_{I}(G^{r}-G^{a})+G^{r}\right]\delta_{IJ}=\rho_{(0),I}\left[n_{I}+\frac{D^{*}_{I}}{\gamma_{I}}\right]\delta_{IJ},$
(28)

where the first expression can also be derived from eq. 13, and we used that
$G^{r}_{I}=D_{I}^{-1}$ and $G_{I}^{a}=(D_{I}^{*})^{-1}$ in the right-most
expression. The anti-time-ordered Green’s function is

$G_{(0),IJ}^{\bar{t}}=\left[n_{I}(G^{r}-G^{a})-G^{a}\right]\delta_{IJ}=\rho_{(0),I}\left[n_{I}-\frac{D_{I}}{\gamma_{I}}\right]\delta_{IJ}.$
(29)

The propagators at 1st and 2nd order in VIA can be expressed in terms of the
zeroth order propagators by means of the Schwinger-Dyson equation

$\displaystyle G^{ab}_{(1),IJ}$
$\displaystyle=\sum_{c}c\,G^{ac}_{(0),II}(\delta M^{2})_{IJ}G^{cb}_{(0),JJ},$
$\displaystyle G^{ab}_{(2),II}$
$\displaystyle=\sum_{cd}cd\,G^{ac}_{(0),II}(\delta
M^{2})_{IJ}G^{cd}_{(0),JJ}(\delta M^{2})_{JI}G^{db}_{(0),II},$ (30)

with $J\neq I$. The first and second order propagators are flavor-off-diagonal
and -diagonal respectively.

####  3.4.2 Kinetic equation

For notational convenience we will drop the (0) subscript on the zeroth order
Green’s functions and spectral function in this subsection. The higher order
Green’s functions are labeled by a subscript, and there should thus be no
source of confusion.

At zeroth order the source term vanishes as the zeroth order propagators are
diagonal and therefore the commutator with the mass vanishes. As the zeroth
order propagators satisfy the KMS relation the collision term also vanishes.
At first order the diagonal entries still vanish by virtue of the KMS
relations for the zeroth order propagators. To show that the off-diagonal
source vanishes as well is a bit more non-trivial, but straightforward. Here
we focus on the diagonal source terms at 2nd order, which is the usual order
at which the VIA source is derived, given by eq. 19

$\displaystyle\bar{S}^{(2)}$ $\displaystyle=[\delta
M^{2},(G_{(1)}^{>}+G_{(1)}^{<})]+[M_{d}^{2},(G_{(2)}^{>}+G_{(2)}^{<})]+[\Pi^{>}+\Pi^{<},G^{h}_{(2)}]$
$\displaystyle+\left(\\{\Pi^{>},G_{(2)}^{<}\\}-\\{\Pi^{<},G_{(2)}^{>}\\}\right).$
(31)

Only the first and last terms on the right-hand-side contribute to the
diagonal source term $(\bar{S}^{(2)})_{II}$. The higher order Green’s
functions can be expressed in terms of the zeroth order Green’s functions
using eq. 30.

The contribution of the first term in eq. 31 to the $LL$-component of the
source is

$\displaystyle\bar{S}^{(2)}_{M,LL}$ $\displaystyle\equiv[\delta
M^{2},(G_{(1)}^{>}+G_{(1)}^{<})]=m^{2}_{LR}(G^{>}_{(1),RL}+G^{<}_{(1),RL})-(G^{>}_{(1),LR}+G^{<}_{(1),LR})m^{2}_{RL}$
$\displaystyle=|m_{LR}|^{4}\left((G^{>}-G^{<})_{RR}(G^{t}+G^{\bar{t}})_{LL}-(G^{>}-G^{<})_{LL}(G^{t}+G^{\bar{t}})_{RR}\right)$
$\displaystyle=|m_{LR}|^{4}\rho_{L}\rho_{R}\left[(2n_{L}-1)-(2n_{R}-1)\right]=2|m_{LR}|^{4}\rho_{L}\rho_{R}(n_{L}-n_{R}),$
(32)

where we used that $(G^{t}+G^{\bar{t}})_{II}=(2n_{I}-1)\rho_{I}$ and
$(G^{>}-G^{<})_{II}=\rho_{I}$. The collision term contributes

$\displaystyle\bar{S}^{(2)}_{C,LL}$
$\displaystyle\equiv\left(\\{\Pi^{>},G_{(2)}^{<}\\}-\\{\Pi^{<},G_{(2)}^{>}\\}\right)_{LL}\
$ (33)
$\displaystyle=2|m_{LR}|^{4}\,\Big{[}(\Pi^{>}_{LL}G^{<}_{LL}-G^{>}_{LL}\Pi^{<}_{LL})(G^{\bar{t}}_{LL}G^{\bar{t}}_{RR}+G^{t}_{LL}G^{t}_{RR})$
$\displaystyle\hskip
42.67912pt+(\Pi^{<}_{LL}G^{>}_{LL}G^{<}_{RR}G^{>}_{LL}-\Pi^{>}_{LL}G^{<}_{LL}G^{>}_{RR}G^{<}_{LL})+(\Pi^{<}_{LL}G^{>}_{RR}-\Pi^{>}_{LL}G^{<}_{RR})G^{\bar{t}}_{LL}G^{t}_{LL}\Big{]}.$

The term on the 2nd line vanishes by virtue of the KMS relation for the zeroth
order propagator and self-energy. The first term on the last line becomes
becomes

$\displaystyle{\rm term}_{1}$
$\displaystyle\equiv(\Pi^{<}_{LL}G^{>}_{LL}G^{<}_{RR}G^{>}_{LL}-\Pi^{>}_{LL}G^{<}_{LL}G^{>}_{RR}G^{<}_{LL})$
$\displaystyle=\gamma_{L}\rho_{L}^{2}\rho_{R}n_{L}(1+n_{L})\left(n_{R}(1+n_{L})-n_{L}(1+n_{R})\right)$
$\displaystyle=-\gamma_{L}\rho_{L}^{2}\rho_{R}n_{L}(1+n_{L})(n_{L}-n_{R}).$
(34)

The second term on the last line equals

$\displaystyle{\rm term}_{2}$
$\displaystyle\equiv(\Pi^{<}_{LL}G^{>}_{RR}-\Pi^{>}_{LL}G^{<}_{RR})G^{\bar{t}}_{LL}G^{t}_{LL}$
$\displaystyle=\gamma_{L}\rho_{L}^{2}\rho_{R}\left(n_{L}-n_{R}\right)\left(n_{L}+\frac{D^{*}_{L}}{\gamma_{L}}\right)\left(n_{L}-\frac{D_{L}}{\gamma_{L}}\right).$
(35)

Adding them up we find

$\displaystyle({\rm term}_{1}+{\rm term}_{2})$
$\displaystyle=-\bar{\gamma}_{L}(\rho_{L}^{+})^{2}\rho_{R}^{-}(n_{L}-n_{R})\left[n_{L}(1-n_{L})+\left(n_{L}+\frac{D^{*}_{L,+}}{\bar{\gamma}_{L}}\right)\left(n_{L}-\frac{D_{L,+}}{\bar{\gamma}_{L}}\right)\right]$
$\displaystyle=-\gamma_{L}\rho_{L}^{2}\rho_{R}(n_{L}-n_{R})\frac{D^{*}_{L}D_{L}}{\gamma_{L}^{2}}=-\rho_{L}\rho_{R}(n_{L}-n_{R}),$
(36)

where we used that $D_{L}-D^{*}_{L}+\gamma_{L}=0$. The final result is then

$\displaystyle\bar{S}_{LL}^{(2)}$
$\displaystyle=\bar{S}^{(2)}_{M,LL}+\bar{S}^{(2)}_{C,LL}=2|m_{LR}|^{4}\rho_{L}\rho_{R}(n_{L}-n_{R})-2|m_{LR}|^{2}\rho_{L}\rho_{R}(n_{L}-n_{R})=0.$

The VIA source thus cancels to 2nd order in the VIA expansion. Likewise, the
right-handed source term and all off-diagonal contributions to the source term
vanish.

###  3.5 Comparison with usual derivation of the VIA source

Let’s compare our results with the standard derivation of the VIA source found
in the literature Riotto:1995hh ; Lee:2004we . In this approach the self-
energy is expanded to 2nd order

$\Pi^{\lambda}_{II,(2)}=-(\delta M^{2})_{IJ}G^{\lambda}_{JJ}(\delta
M^{2})_{JI},$ (37)

and Wightman functions are taken at leading order. The diagonal source
originates from the collision term as all other terms in the kinetic equation
vanish

$\displaystyle\bar{S}_{LL}^{(2)}\big{|}_{\rm usual}$
$\displaystyle=\\{\Pi_{LL,(2)}^{>},G_{LL}^{<}\\}-\\{\Pi_{(LL),(2)}^{<},G_{LL}^{>}\\}$
$\displaystyle=-|m_{LR}|^{4}\left(\\{G_{RR}^{>},G_{LL}^{<}\\}-\\{G_{RR}^{<},G_{LL}^{>}\\}\right)=-|m_{LR}|^{4}\rho_{L}\rho_{R}(n_{L}-n_{R}).$
(38)

This is the same expression for the collision term as in our approach derived
above eq. 36. To bring this into more standard VIA notation we perform an
inverse Wigner transform on the first line in eq. 38 above to evaluate the
source eq. 19 in position space:

$\displaystyle S_{LL}^{(2)}(x)$ $\displaystyle=$ $\displaystyle-2\int
d^{4}y{\rm
Re}\left[m_{LR}^{2}(x)G_{RR}^{<}(x,y)m_{RL}^{2}(y)G_{LL}^{>}(y,x)\right.$
$\displaystyle\left.\hskip 55.48277pt-
m_{RL}^{2}(x)G_{RR}^{>}x,y)m_{LR}^{2}(y)G_{LL}^{<}(y,x)\right]$
$\displaystyle=$ $\displaystyle-2\int d^{4}y\,\Big{(}{\rm
Re}\left[g(x,y)+g(y,x)\right]{\rm
Re}\left[G_{RR}^{<}(x,y)G_{LL}^{>}(y,x)-G_{RR}^{>}(x,y)G^{<}_{LL}(y,x)\right]$
$\displaystyle\hskip 45.52458pt+{\rm Im}\left[g(x,y)+g(y,x)\right]{\rm
Im}\left[G_{RR}^{<}(x,y)G_{LL}^{>}(y,x)-G_{RR}^{>}(x,y)G^{<}_{LL}(y,x)\right]\Big{)},$

with $g(x,y)=m_{LR}^{2}(x)m_{RL}^{2}(y)$, and where for notational convenience
we have dropped the ‘usual’ subscript. Now expand the masses in the integrand

$m_{IJ}^{2}(y)=m_{IJ}^{2}(x)+(x-y)^{\mu}\partial_{\mu}m^{2}_{IJ}(x)+{\cal
O}(\partial^{2}).$ (40)

Although this expansion also assumes a slowly varying bubble wall background,
it is different from the derivative expansion, i.e. the expansion in terms of
the diamond operator, of the KB equations. The leading order term of this mass
expansion gives a CP-conserving source term, whereas from the next-to-leading
term we recover the usual CP-violating (CPV) source term found in the
literature:

$\displaystyle S^{(2)}_{\rm CPV}=2{\rm
Im}\left[m_{LR}^{2}\partial_{\mu}m^{2}_{RL}\right]\int d^{4}y\,(y-x)^{\mu}$
$\displaystyle\Big{(}G_{RR}^{<}(x,y)G_{LL}^{>}(y,x)$
$\displaystyle\qquad-G^{>}_{RR}(x,y)G_{LL}^{<}(y,x)\Big{)}.$ (41)

Hence, it was concluded that the VIA source term is non-zero. We believe that
this erroneous result is a consequence of the non-1PI approach, which is not a
systematic perturbative expansion, and obscures the correct counting of
diagrams. Indeed, our more systematic approach of the previous subsections
shows that if all contributions are carefully taken into account the source
term cancels.

In Postma:2019scv  the self-energy was calculated to next-to-leading order,
that is, to 4th order in the VIA expansion. This calculation thus likewise
works with non 1PI-diagrams, and thus suffers from the same issues as the 2nd
order calculation. As we will see in the next subsection, the VIA source
cancels to all orders in the vev-insertion expansion.

###  3.6 Resummed Green’s functions and source term

With the thermal equilibrium approximation for the thermal energies in eq. 25,
the KB equations at leading order in the derivative expansion can be solved
exactly for the off-diagonal mass matrix, and there is no need to actually
perform a vev-insertion expansion. The equations to solve are

$\displaystyle k^{2}G^{\lambda}$
$\displaystyle=\frac{1}{2}\left(\\{M^{2},G^{\lambda}\\}+\\{\Pi^{\lambda},G^{h}\\}+\frac{1}{2}([\Pi^{>},G^{<}]-[\Pi^{<},G^{>}])\right),$
$\displaystyle k^{2}G^{t}$
$\displaystyle=1+\frac{1}{2}\left(\\{M^{2}+\Pi^{t}-\Pi^{h},G^{t}\\}-\Pi^{<}G^{>}-G^{<}\Pi^{>}\right),$
(42)

with thermal self-energies given in eq. 25. The solutions for the Wightman
functions are

$\displaystyle G^{\lambda}_{LL}$
$\displaystyle=\frac{1}{\mathcal{D}_{+}\mathcal{D}_{-}}\frac{\gamma_{R}\gamma_{L}}{\rho_{R}}\left(g^{\lambda}_{L}+g^{\lambda}_{R}\frac{\rho_{R}}{\gamma_{L}}|m_{LR}|^{4}\right),$
$\displaystyle G^{\lambda}_{LR}$
$\displaystyle=\frac{m_{LR}^{2}}{\mathcal{D}_{+}\mathcal{D}_{-}}\left(\gamma_{R}g_{R}^{\lambda}(k^{2}-m_{L}^{2})+\gamma_{L}g^{\lambda}_{L}(k^{2}-m_{R}^{2})+\frac{1}{2}\gamma_{R}\gamma_{L}(g^{\lambda}_{R}-g^{\lambda}_{L})\right),$
(43)

with $g^{\lambda}_{I}$ defined in eq. 4, $\rho_{L},\rho_{R}$ the zeroth order
spectral function eq. 27, and

$\mathcal{D}_{\pm}=(k^{2}-m_{L}^{2}\pm\gamma_{L}/2)(k^{2}-m_{R}^{2}\pm\gamma_{R}/2)-|m_{LR}|^{2}.$
(44)

The other flavor components can be found through
$G^{\lambda}_{RR}=G^{\lambda}_{LL}|_{L\leftrightarrow R}$ and
$G^{\lambda}_{RL}=G^{\lambda}_{LR}|_{L\leftrightarrow R}$. Note that

$\lim_{m^{2}_{LR},m^{2}_{RL}\to
0}\left\\{\frac{1}{\mathcal{D}^{+}\mathcal{D}^{-}}\frac{\gamma_{R}\gamma_{L}}{\rho_{R}},\,\frac{1}{\mathcal{D}^{+}\mathcal{D}^{-}}\frac{\gamma_{R}\gamma_{L}}{\rho_{L}}\right\\}=\\{\rho_{L},\,\rho_{R}\\},$
(45)

and we retrieve the perturbative zeroth order Green’s function in this limit.

Note that the denominators of the resummed propagators are shifted by the off-
diagonal mass squared $|m_{LR}|^{2}$. Consequently, the separate contributions
from the collision term and from the mass commutator term do not suffer from a
pinch singularity, as foreseen in Kainulainen:2021oqs , as the divergence in
the limit of equal hermitian self-energies $\Pi^{h}_{LL}=\Pi^{h}_{RR}$ and
vanishing decay widths $\gamma_{I}\to 0$ is cut off.

To evaluate the off-diagonal source term one also needs the (anti) time-
ordered Green’s function

$\displaystyle G^{t}_{LR}$
$\displaystyle=\frac{m_{LR}^{2}}{\cal{D}_{+}\cal{D}_{-}}\Big{(}(k^{2}-m_{L}^{2})\gamma_{R}(n_{R}+\frac{1}{2})+(k^{2}-m_{R}^{2})\gamma_{L}(n_{L}+\frac{1}{2})-|m_{LR}|^{2}$
$\displaystyle\hskip
56.9055pt+\frac{1}{4}\gamma_{L}\gamma_{R}(2n_{R}-2n_{L}+1)+(k^{2}-m_{R}^{2})(k^{2}-m_{L}^{2})\Big{)},$
(46)

and $G^{\bar{t}}_{LR}=-G^{t}_{LR}|_{\gamma_{I}\to-\gamma_{I}}$.

Using the explicit solutions, the source term eq. 19

$\displaystyle\bar{S}$
$\displaystyle=[M^{2},(G^{>}+G^{<})]+[\Pi^{>}+\Pi^{<},G^{h}]+\left(\\{\Pi^{>},G^{<}\\}-\\{\Pi^{<},G^{>}\\}\right)=0,$
(47)

vanishes identically. For the flavor-diagonal source terms, this is due to a
cancellation between the commutator with the off-diagonal mass $\delta M^{2}$
and the collision term. Explicitly, using eq. 43, we find

$\displaystyle[\delta M^{2},(G^{>}+G^{<})]_{LL}$
$\displaystyle=2|m_{LR}|^{4}\frac{\gamma_{L}\gamma_{R}(n_{L}-n_{R})}{\mathcal{D}_{+}\mathcal{D}_{-}},$
$\displaystyle\\{\Pi^{>},G^{<}\\}_{LL}-\\{\Pi^{<},G^{>}\\}_{LL}$
$\displaystyle=-2|m_{LR}|^{4}\frac{\gamma_{L}\gamma_{R}(n_{L}-n_{R})}{\mathcal{D}_{+}\mathcal{D}_{-}}.$
(48)

And thus $\bar{S}_{LL}=0$ exactly, to all orders in the vev-insertion
expansion.

###  3.7 Discussion: flavor dynamics

The thermal corrections arise from model-dependent interactions with the bath
that are (approximately) described by thermal equilibrium physics. The flavor
physics is only non-trivial when the interaction and mass bases are not
aligned. The standard VIA approach generates a source term for the two flavors
that is equal but opposite in sign, and thus depends crucially on non-trivial
flavor physics. Our choice of the self-energy eq. 25 yields at leading order
in the VIA expansion a Wightman function
$G_{II}^{\lambda}\propto\Pi_{II}^{\lambda}\propto g^{\lambda}_{II}$ as in eq.
26, which is of the usual form used in the VIA literature Riotto:1995hh ;
Riotto:1998zb . In this work it is our aim to show that, following the
assumptions made in the literature, the VIA source cancels at leading order in
the derivative expansion. We note, though, that the cancellation of the source
is independent of the specific choice of self-energy. This can be checked
explicitly by solving the KB equations for generic (flavor-diagonal)
$\Pi^{\lambda}_{II}$.

At leading order in the gradient expansion the individual terms in the kinetic
equation are proportional to $(n_{L}-n_{R})$. Although the source term cancels
at this order in the derivative expansion, a non-zero source may still arise
at higher order. This would require non-trivial flavor dynamics, different
interactions for the flavor states, to allow for a symmetry $n_{L}\neq n_{R}$
to build up. This is consistent with Ref. Kainulainen:2021oqs , who did not
find a VIA source at any order of the derivative expansion for fermions with
vector-like bath interactions (i.e. no flavour dynamics). The fermionic case
is discussed in the next section.

Lastly, we note that the propagators at higher order in the VIA expansion eq.
30 and the fully resummed propagators eq. 43 no longer satisfy the KMS
relations as in eq. 26 with the spectral function derived at the same order in
VIA. The reason is that at lowest order in VIA the mass and flavor eigenstates
still coincide, but at higher order this is no longer the case. Whereas the
spectral function gives information about the mass eigenstates, the KMS
relations for the self-energies are associated to the number densities of the
flavor eigenstates. This is motivated by the rapid interactions with the bath,
and to a good approximation the bath can be regarded as a collection of
particles in flavor eigenstates. Had $\delta M^{2}$ instead been flavor-
diagonal, and thus flavor and mass eigenstates aligned at all orders in VIA,
then one would have found a KMS relation eq. 26 for the Wightman functions at
all orders. The collision term in the kinetic equations vanishes for Wightman
functions and self-energies that satisfy the KMS relations. Since this is not
the case at higher order in VIA, the collision term will contribute to the
source.

##  4 Fermions

Although the constraint and kinetic equations for bosons eq. 8 and fermions
eq. 9 have a very similar structure, the spinor structure of the latter seems
to complicate the analysis considerably. For example, the structure of the
fermionic thermal self-energy is such that it leads to the appearance of
collective plasma modes, see Appendix B. Fortunately, we do not have to delve
into these complications for the derivation of the VIA source. In fact, as we
will see, we can expand both the self-energies and Green’s functions in
helicity eigenstates. The spinor structure then drops out of the KB equations,
and the derivation and cancellation of the VIA source is then in almost one-
to-one correspondence to the bosonic case.

With a slight abuse of terminology, we will now refer to the two chiralities
as different ‘flavors’, because the chiralities have different interactions
with the thermal plasma in a chiral theory such as the SM. Once thermal
corrections are included, the propagating and interaction states are not
aligned, which is essential for the VIA source.

###  4.1 Lagrangian

For definiteness, consider two Weyl fermions with left and right chirality
coupled via a background-dependent Dirac mass $M_{0}=M_{0}(v)$

$\displaystyle{\cal L}$
$\displaystyle=i\bar{\psi}\not{\partial}\psi-\bar{\psi}M_{0}\psi=i\bar{\psi}\not{\partial}\psi-\bar{\psi}_{L}m_{LR}\psi_{R}-\bar{\psi}_{R}m_{RL}\psi_{L},$
(49)

with $m_{LR}=(m_{RL})^{*}$, $P_{R}M_{0}P_{R}=m_{LR}$ and
$P_{L}M_{0}P_{L}=m_{RL}$. The vev insertion approximation consists of treating
$m_{LR}=(m_{RL})^{*}$ as a perturbation, and expanding the KB-equations in
this small quantity.

###  4.2 KB equations

The KB equations eq. 9 for the Wightman functions $G^{\lambda}$ and time-
ordered propagator $G^{t}$ are

$\displaystyle\\{\not{k},G^{\lambda}\\}$ $\displaystyle={\rm
e}^{-i\diamond}\left(\\{M,G^{\lambda}\\}+\\{\Pi^{\lambda},G^{h}\\}+\frac{1}{2}([\Pi^{>},G^{<}]-[\Pi^{<},G^{>}])\right),$
$\displaystyle\frac{1}{2}\\{\not{k},G^{t}\\}$ $\displaystyle=1+\frac{1}{2}{\rm
e}^{-i\diamond}\left(\\{M+\Pi^{t}-\Pi^{h},G^{t}\\}-\Pi^{<}G^{>}-G^{<}\Pi^{>}\right),$
(50)

where we defined $M=M_{0}+\Pi^{h}$ with $M_{0}$ the tree-level Dirac mass
matrix, and $\Pi^{h}$ the thermal mass. The source terms can be derived from
the kinetic equation for the Wightman functions

$\displaystyle\frac{i}{2}\\{\not{\partial}_{x},G^{\lambda}\\}$
$\displaystyle={\rm
e}^{-i\diamond}\left([M,G^{\lambda}]+[\Pi^{\lambda},G^{h}]+\frac{1}{2}\left(\\{\Pi^{>},G^{<}\\}-\\{\Pi^{<},G^{>}\\}\right)\right).$
(51)

Integrating the left-hand side over 4-momentum gives Riotto:1997vy

$\displaystyle{\rm Tr}\left[\int\frac{{\rm
d}^{4}k}{(2\pi)^{4}}i\\{\not{\partial}_{x},G^{\lambda}(k,x)\\}\right]$
$\displaystyle=\lim_{r\to 0}{\rm
Tr}\,\left[i(\not{\partial}_{x}G^{\lambda}(r,x)+G^{\lambda}(r,x)\overleftarrow{\not{\partial}}_{x})\right]$
$\displaystyle=\partial_{x^{\mu}}\langle\bar{\psi}(x)\gamma^{\mu}\psi(x)\rangle=\partial_{\mu}\langle
J^{\mu}\rangle,$ (52)

with ${\rm Tr}$ the trace over Dirac spinor indices. The position-space limit
$r\to 0$ corresponds to taking the leading order derivative expansion in
Wigner space, i.e. setting ${\rm e}^{-i\diamond}=1$, which is picked out by
the integration over 4-momenta. The evolution of the left-handed current is
then defined as

$\displaystyle\partial_{\mu}\langle
J_{L}^{\mu}\rangle\equiv\partial_{\mu}\langle\bar{\psi}_{L}(x)\gamma^{\mu}\psi_{L}(x)\rangle={\rm
Tr}\Big{[}\int\frac{{\rm d}^{4}k}{(2\pi)^{4}}i\Big{(}$ $\displaystyle
P_{R}\not{\partial}_{x}P_{L}G^{\lambda}(k,x)P_{R}$
$\displaystyle\qquad+P_{L}G^{\lambda}(k,x)P_{R}\not{\partial}_{x}P_{L}\Big{)}\Big{]}.$
(53)

Hence, the source term for the left-handed fields can be written as

$\displaystyle S_{LL}$ $\displaystyle={\rm Tr}\Big{[}\int\frac{{\rm
d}^{4}k}{(2\pi)^{4}}\left([M,G^{>}+G^{<}]+[\Pi^{>}+\Pi^{<},G^{h}]+\left(\\{\Pi^{>},G^{<}\\}-\\{\Pi^{<},G^{>}\\}\right)\right)_{LL}\Big{]}$
$\displaystyle\equiv\int\frac{{\rm d}^{4}k}{(2\pi)^{4}}\bar{S}_{LL},$ (54)

with $(AB)_{LL}=\sum_{I=L,R}A_{LI}B_{IL}$. Note in this respect that for the
propagator $G_{LL}=P_{L}GP_{R}$ while for the mass/self-energy
$M_{LL}=P_{R}MP_{L},\,\Pi_{LL}=P_{R}\Pi P_{L}$. In the last line we defined
the integrand $\bar{S}_{LL}$ for future reference.

###  4.3 Thermal corrections

Let’s first consider massless fermions and set the Dirac mass to zero. This
would be the starting point in the vev-insertion approximation, where at
leading order the off-diagonal vev-dependent mass terms are set to zero. The
self-energies are flavor-diagonal, and we approximate them with the thermal
self-energies that satisfy the KMS relations eq. 12. We can then asses the
effects of the self-energies in the diagonal basis of retarded and advanced
propagators using eq. 13, where we work at leading order in the derivative
expansion.

The self-energies for the left and right chiralities may differ in a chiral
theory. For a massless fermion the hermitian and anti-hermitian self-energy
can be written as

$\displaystyle\Pi^{h}$
$\displaystyle=P_{R}\left(a_{L}\not{k}+b_{L}\not{u}\right)P_{L}+P_{L}\left(a_{R}\not{k}+b_{R}\not{u}\right)P_{R},$
$\displaystyle\Pi^{\cal A}$
$\displaystyle=P_{R}\left(2\bar{\Gamma}_{L}\gamma_{0}\right)P_{L}+P_{L}\left(2\bar{\Gamma}_{R}\gamma_{0}\right)P_{R},$
(55)

with $a_{I},b_{I}$ Lorentz invariants and functions of $k=|\mathbf{k}|$. In
vacuum $b_{I}=0$ by Lorentz invariance, but at finite temperature the center
of mass frame of the plasma introduces a special Lorentz frame. In the plasma
rest frame the plasma four velocity is $u^{\mu}=(1,0,0,0)$ and
$\not{u}=\gamma^{0}$. The anti-hermitian self-energy is calculated in the
zero-momentum limit Braaten:1992gd , and can then be related to the decay
rate. Away from this limit, the expression is likely of the same general form
as $\Pi^{h}$, but for simplicity we will stick to the simpler expression for
$\Pi^{\cal A}$ in eq. 55.

It will be useful to work in the basis of 2-dimensional Weyl spinors, and we
define

$\displaystyle M_{0}=\left(\begin{array}[]{cc}m_{RL}&0\\\
0&m_{LR}\end{array}\right),\quad\Pi^{ab}=\left(\begin{array}[]{cc}0&\Pi^{ab}_{RR}\\\
\Pi^{ab}_{LL}&0\end{array}\right),\quad
G^{ab}=\left(\begin{array}[]{cc}G_{LR}^{ab}&G^{ab}_{LL}\\\
G^{ab}_{RR}&G_{RL}^{ab}\end{array}\right),$ (62)

with the full matrices $M_{0},\Pi,G$ all 4-by-4 in Dirac spinor space, and the
sub-blocks $\Pi_{IJ},G_{IJ},M_{IJ}$ 2-by-2 matrices in Weyl spinor space. Note
that this is a change of notation, as below eq. 54 the same notation was used
for the projection of propagators and self-energies in Dirac space. From now
on, all propagators, self-energies and masses will be given in Weyl space, and
we do not expect this to lead to confusion.

We introduce the helicity projection operators

$P^{\pm}=\frac{1}{2}(1\pm\boldsymbol{\sigma}\cdot\hat{k}),$ (63)

which satisfy $P^{+}P^{-}=0$, $(P^{\pm})^{2}=P^{\pm}$ and $P^{+}+P^{-}=1$. The
hermitian self-energy can then be written as follows

$\Pi^{h}_{LL}=\Pi^{h}_{LL,+}P^{+}+\Pi^{h}_{LL,-}P^{-},\quad\Pi^{h}_{RR}=\Pi^{h}_{RR,-}P^{+}+\Pi^{h}_{RR,+}P^{-},$
(64)

with $\Pi^{h}_{II,\pm}=a_{I}K_{\pm}+b_{I}$ and $K_{\pm}=k^{0}\pm k$. To
leading order in the derivative expansion the equation for the
retarded/advanced Green’s function is
$\frac{1}{2}\\{\not{k},G^{\alpha}\\}-\frac{1}{2}\\{\Pi^{\alpha},G^{\alpha}\\}=1$.
We rewrite $\not{k}$ in the Weyl basis

$\displaystyle\not{k}$
$\displaystyle=\left(\begin{array}[]{cc}0&k^{0}-\mathbf{k}\cdot\boldsymbol{\sigma}\\\
k^{0}+\mathbf{k}\cdot\boldsymbol{\sigma}&0\end{array}\right)=\left(\begin{array}[]{cc}0&K_{-}P^{+}+K_{+}P^{-}\\\
K_{+}P^{+}+K_{-}P^{-}&0\end{array}\right).$ (69)

The solution for the inverse retarded propagator becomes

$\displaystyle(G_{LL}^{r})^{-1}$
$\displaystyle=(K_{-}-\Pi^{h}_{LL,-}+2i\bar{\Gamma}_{L})P^{-}+(K_{+}-\Pi^{h}_{LL,+}+2i\bar{\Gamma}_{L})P^{+},$
$\displaystyle(G_{RR}^{r})^{-1}$
$\displaystyle=(K_{+}-\Pi^{h}_{RR,+}+2i\bar{\Gamma}_{R})P^{-}+(K_{-}-\Pi^{h}_{RR,-}+2i\bar{\Gamma}_{R})P^{+}.$
(70)

Inverting this relation gives

$G_{LL}^{r}=G_{LL,+}^{r}P^{+}+G_{LL,-}^{r}P^{-},\quad
G_{RR}^{r}=G_{RR,-}^{r}P^{+}+G_{RR,+}^{r}P^{-},$ (71)

with

$G^{r}_{II,\pm}=(K_{\pm}-\Pi^{h}_{II,\pm}+2i\bar{\Gamma}_{I})^{-1}\equiv
D_{I,\pm}^{-1}.$ (72)

The advanced propagator is obtained by taking
$i\bar{\Gamma}_{I}\to-i\bar{\Gamma}_{I}$.

To re-express the results in the basis of Wightman functions, we first derive
the spectral function. Define

$\displaystyle\rho_{I,\pm}=\left(G_{II,\pm}^{r}-G_{II,\pm}^{a}\right)=\frac{1}{D_{I,\pm}}-\frac{1}{D_{I,\pm}^{*}}=\frac{\bar{\gamma}_{I}}{D_{I,\pm}D_{I,\pm}^{*}}=\frac{\bar{\gamma}_{I}}{(K_{\pm}-\Pi^{h}_{II,\pm})^{2}-\bar{\gamma}_{I}^{2}/4},$
(73)

with $\bar{\gamma}_{I}=-4i\bar{\Gamma}_{I}$, where we use the barred notation
to avoid possible confusion with a Dirac matrix. Then
$\rho_{L}=\rho_{L,+}P^{+}+\rho_{L,-}P^{-}$ and
$\rho_{R}=\rho_{R,-}P^{+}+\rho_{R,+}P^{-}$. With this notation we write the
self-energies as

$\Pi_{II}^{\lambda}=\bar{\gamma}_{I}g^{\lambda}_{I},$ (74)

with $g^{\lambda}$ defined in eq. 4. This is analogous to our choice for the
bosonic case eq. 25, and follows the assumptions in the VIA literature. It is
at this point that our treatment diverges from the analysis in Ref.
Kainulainen:2021oqs , which is based on vector-like interactions with the
bath, whereas eq. 74 is expected in a chiral theory. The discussion on flavor
dynamics in section 3.7 carries over to the fermionic case.

The diagonal Green’s functions are expanded in projection operators

$G^{ab}_{LL}=G^{ab}_{LL,+}P^{+}+G^{ab}_{LL,-}P^{-},\quad
G^{ab}_{RR}=G^{ab}_{RR,-}P^{+}+G^{ab}_{RR,+}P^{-}.$ (75)

The Wightman functions are then

$G^{\lambda}_{II,\pm}=g^{\lambda}_{II}\rho_{I}^{\pm}.$ (76)

The (anti) time-ordered propagators
$G_{II}^{t}=-(G^{r}-G^{a})_{II}n_{I}+G^{r}_{II}$ and
$G_{II}^{\bar{t}}=-(G^{r}-G^{a})_{II}n_{I}-G^{a}_{II}$ can be written as

$\displaystyle
G^{t}_{II,\pm}=\left(-n_{I}+\frac{D_{I,\pm}^{*}}{\bar{\gamma}_{I}}\right)\rho_{I}^{\pm},\qquad
G^{\bar{t}}_{II,\pm}=\left(-n_{I}-\frac{D_{I,\pm}}{\bar{\gamma}_{I}}\right)\rho_{I}^{\pm},$
(77)

where we used eq. 73.

The constraint equations at leading order in both the derivative expansion and
the VIA expansion can be solved to directly find the Wightman functions and
time-ordered propagators. This is what we have shown explicitly for bosons in
section 3.4.1. We will not repeat the exercise here for fermions, but it is
straightforward to check that indeed the solutions given above solve the
leading order constraint equations.

To calculate the source term and show that also in the fermionic case there is
a cancellation, we can work with the spectral function given in eq. 73, which
is a solution of the constraint equation at leading order in VIA. Although
there is no need to determine the pole structure and identify the particle and
collective hole modes, this is further discussed in appendix B, to show that
this is indeed the same spectral function as used in the usual VIA approach.

###  4.4 VIA approximation

The propagators found in the previous subsection are the leading order
propagators in the VIA expansion. The KB equations have a similar form as
those for bosons, and it is no surprise that the (leading order) Green’s
functions found in the previous subsection have the same structure as those
for bosons as well. Likewise, as we will show, the cancellation of the source
term derived from the kinetic equation goes along the same lines as what we
have already seen for bosons. In this subsection we will discuss the VIA
approximation, and in the next subsection we derive the source from the
resummed propagators.

The propagators at 1st and 2nd order in VIA can be expressed in terms of these
zeroth order propagators by means of the Schwinger-Dyson eq. 30, which becomes
in Weyl space

$\displaystyle G^{ab}_{(1),LR}$
$\displaystyle=m_{LR}\,\sum_{c}c\left(G^{ac}_{LL,+}G^{cb}_{RR,-}P_{+}+G^{ac}_{LL,-}G^{cb}_{RR,+}P_{-}\right),$
$\displaystyle G^{ab}_{(2),LL}$
$\displaystyle=|m_{LR}|^{2}\sum_{cd}cd\left(G^{ac}_{LL,+}G^{cd}_{RR,-}G^{db}_{LL,+}P_{+}+G^{ac}_{LL,-}G^{cd}_{RR,+}G^{db}_{LL,-}P_{-}\right).$
(78)

In this subsection, all propagators without a $(..)$ subscript are again to be
understood as the zeroth order propagator.

We can then evaluate the source term in the VIA expansion. Let’s focus on the
left-handed source eq. 54 at 2nd order in VIA. We insert $1=(P^{-}+P^{+})$ in
the trace, and denote the projections of the source onto $P^{\pm}$ by
$\bar{S}^{\pm}$. We concentrate here on $\bar{S}_{LL}^{+}$, but it is
straightforward to check that all other diagonal source contributions
$S_{II}^{\pm}$ vanish as well. At 2nd order in VIA,

$\displaystyle\bar{S}_{LL}^{(2)+}$ $\displaystyle={\rm
Tr}\left[P^{+}\left([M_{0},G_{(1)}^{>}+G_{(1)}^{<}]+\left(\\{\Pi^{>},G_{(2)}^{<}\\}-\\{\Pi^{<},G_{(2)}^{>}\\}\right)\right)_{LL}\right],$
(79)

with now ${\rm Tr}$ denoting the trace in Weyl space, and
$(AB)_{LL}=\sum_{I=L,R}A_{LI}B_{IL}$ in Weyl space notation. $M_{0}$ is the
tree-level (Dirac) mass.

Consider first the commutator term

$\displaystyle\bar{S}^{(2)+}_{M,LL}$ $\displaystyle\equiv{\rm
Tr}\left[P^{+}[M_{0},G_{(1)}^{\lambda}]\right]={\rm
Tr}\left[m_{LR}P^{+}(G^{>}_{(1),RL}+G^{<}_{(1),RL})-P^{+}(G^{>}_{(1),LR}+G^{<}_{(1),LR})m_{RL}\right]$
$\displaystyle={\rm
Tr}[P^{+}]|m_{LR}|^{2}\left[(G^{t}+G^{\bar{t}})_{LL,+}(G^{>}-G^{<})_{RR,-}-(G^{t}+G^{\bar{t}})_{RR,-}(G^{>}-G^{<})_{LL,+}\right]$
$\displaystyle=-2|m_{LR}|^{2}\rho_{L}^{+}\rho_{R}^{-}(n_{L}-n_{R}),$ (80)

where we used that
$(G^{t}_{\pm}+G^{\bar{t}}_{\pm})_{II}=(1-2n_{I})\rho_{I}^{\pm}$ and
$(G^{>}-G^{<})_{\pm,II}=\rho_{I}^{\pm}$. Further ${\rm Tr}(P^{+})=1$. This has
exactly the same structure as in the bosonic case eq. 32, except for an overal
sign difference because of the difference between bose and fermi statistics.

The collision term contributes

$\displaystyle\bar{S}^{(2)+}_{C,LL}$ $\displaystyle\equiv 2\,{\rm
Tr}\left[P^{+}\left(G_{(2),LL}^{<}\Pi_{LL}^{>}-G_{(2),LL}^{>}\Pi_{LL}^{<}\right)\right]=2|m_{LR}|^{2}\,({\rm
term}_{1}+{\rm term}_{2}),$ (81)

where we have used the KMS relations to already cancel some contributions,
fully analogous to what happens in the bosonic expression eq. 33. The two
remaining terms are then

$\displaystyle{\rm term}_{1}$ $\displaystyle\equiv{\rm
Tr}\left[P^{+}\left(\Pi^{<}_{LL}G^{>}_{LL}G^{<}_{RR}G^{>}_{LL}-\Pi^{>}_{LL}G^{<}_{LL}G^{>}_{RR}G^{<}_{LL}\right)\right]$
$\displaystyle={\rm
Tr}[P^{+}](\Pi^{<}_{LL}G^{>}_{+,LL}G^{<}_{-,RR}G^{>}_{+,LL}-\Pi^{>}_{LL}G^{<}_{+,LL}G^{>}_{-,RR}G^{<}_{+,LL})$
$\displaystyle=-\bar{\gamma}_{L}(\rho_{L}^{+})^{2}\rho_{R}^{-}n_{L}(1-n_{L})(n_{L}-n_{R}),$
(82)

and

$\displaystyle{\rm term}_{2}$ $\displaystyle\equiv{\rm
Tr}\left[P^{+}(\Pi^{<}_{LL}G^{>}_{RR}-\Pi^{>}_{LL}G^{<}_{RR})G^{\bar{t}}_{LL}G^{t}_{LL}\right]=(\Pi^{<}_{LL}G^{>}_{-,RR}-\Pi^{>}_{LL}G^{<}_{-,RR})G^{\bar{t}}_{+,LL}G^{t}_{+,LL}$
$\displaystyle=-\bar{\gamma}_{L}(\rho_{L}^{+})^{2}\rho_{R}^{-}\left(n_{L}-\frac{D^{*}_{L,+}}{\bar{\gamma}_{L}}\right)\left(n_{L}+\frac{D_{L,+}}{\bar{\gamma}_{L}}\right)\left(n_{L}-n_{R}\right).$
(83)

Just as for bosons, the result simplifies when we add the two terms

$\displaystyle({\rm term}_{1}+{\rm term}_{1})$
$\displaystyle=\rho_{L}^{+}\rho_{R}^{-}(n_{L}-n_{R})$ (84)

where we used that $D_{L,+}-D^{*}_{L,+}+\bar{\gamma}_{L}=0$. The final result
is then

$\displaystyle\bar{S}^{(2)+}$
$\displaystyle=\bar{S}^{(2)+}_{M,LL}+\bar{S}^{(2)+}_{C,LL}=-2|m_{LR}|^{2}\rho_{L}^{+}\rho_{R}^{-}(n_{L}-n_{R})+2|m_{LR}|^{2}\rho_{L}^{+}\rho_{R}^{-}(n_{L}-n_{R})$
$\displaystyle=0.$ (85)

We can likewise calculate the source term projected onto negative helicity
using the $P^{-}$ operator. This amounts to interchanging the plus and minus
labels, and thus also gives a vanishing source term. We conclude that the 2nd
order VIA source cancels also for fermions. In the next subsection we will
show that this cancellation occurs at all orders in VIA.

We can again compare with the usual derivation of the fermionic source term in
the literature Riotto:1995hh ; Lee:2004we , which is derived expanding the
self-energy to 2nd order in VIA. In this approach only the collision term
contributes, and the contribution is equal to our result for the collision
term. To write that contribution to the source in a more familiar form,
perform an inverse Wigner transform on eq. 81 and expand the masses in the
integrand to linear order, just as we did in section 3.4.2 for the bosonic
case. However, as noted before, this standard non-IPI approach misses the
cancellation of the source term.

###  4.5 Resummed Green’s functions and source term

We will now derive the fully resummed source term. The KB constraint equations
at leading derivative order are

$\displaystyle\\{\not{k},G^{\lambda}\\}$
$\displaystyle=\\{M,G^{\lambda}\\}+\\{\Pi^{\lambda},G^{h}\\}+\frac{1}{2}([\Pi^{>},G^{<}]-[\Pi^{<},G^{>}]),$
$\displaystyle\frac{1}{2}\\{\not{k},G^{t}\\}$
$\displaystyle=1+\frac{1}{2}\left(\\{M+\Pi^{t}-\Pi^{h},G^{t}\\}-\Pi^{<}G^{>}-G^{<}\Pi^{>}\right).$
(86)

We express all propagators and self-energies in the Weyl basis as before, see
Appendix A for explicit expressions.

We then project all four components in Weyl space onto the $P^{+}$ and $P^{-}$
propagators, which gives two decoupled systems of equations. The solutions for
$G_{LL,+}$, $G_{LR,+}$ and $G_{RL,-}$ are

$\displaystyle G_{LL,+}^{\lambda}$
$\displaystyle=\frac{1}{\mathcal{D}^{+}\mathcal{D}^{-}}\frac{\bar{\gamma}_{R}\bar{\gamma}_{L}}{\rho^{-}_{R}}\left(g_{L}^{\lambda}+g_{R}^{\lambda}\frac{\rho^{-}_{R}}{\bar{\gamma}_{L}}|m_{LR}|^{2}\right),$
$\displaystyle G_{LR,+}^{\lambda}$
$\displaystyle=\frac{m_{LR}}{\mathcal{D}^{+}\mathcal{D}^{-}}\left(\bar{\gamma}_{R}g_{R}^{\lambda}(K_{+}-\Pi^{h}_{LL,+})+\bar{\gamma}_{L}g_{L}^{\lambda}(K_{-}-\Pi^{h}_{RR,-})+\frac{1}{2}\bar{\gamma}_{L}\bar{\gamma}_{R}(g^{\lambda}_{R}-g^{\lambda}_{L})\right),$
$\displaystyle G_{RL,-}^{\lambda}$
$\displaystyle=\frac{m_{RL}}{\mathcal{D}^{+}\mathcal{D}^{-}}\left(\bar{\gamma}_{R}g_{R}^{\lambda}(K_{+}-\Pi^{h}_{LL,+})+\bar{\gamma}_{L}g_{L}^{\lambda}(K_{-}-\Pi^{h}_{RR,-})+\frac{1}{2}\bar{\gamma}_{L}\bar{\gamma}_{R}(g^{\lambda}_{L}-g^{\lambda}_{R})\right),$
(87)

with $\rho^{\pm}_{I}$ the zeroth order spectral functions eq. 73, and

$\displaystyle\mathcal{D}^{\pm}=(K_{+}-\Pi^{h}_{LL,+}\pm\bar{\gamma}_{L}/2)(K_{-}-\Pi^{h}_{RR,-}\pm\bar{\gamma}_{R}/2)-|m_{LR}|^{2}.$
(88)

In the limit of vanishing Dirac masses

$\lim_{m_{LR},m_{RL}\to
0}\left\\{\frac{1}{D^{+}D^{-}}\frac{\bar{\gamma}_{R}\bar{\gamma}_{L}}{\rho^{-}_{R}},\,\frac{1}{\mathcal{D}^{+}\mathcal{D}^{-}}\frac{\bar{\gamma}_{R}\bar{\gamma}_{L}}{\rho^{+}_{L}}\right\\}=\\{\rho_{L}^{+},\,\rho_{R}^{-}\\},$
(89)

and the 0th order Wightman functions are retrieved. The fermionic solutions
have a similar structure as their bosonic counterpart eq. 43. Pinch
singularities in the separate mass commutator and collision term are again
avoided, as the denominators in the propagator are shifted by the non-zero
Dirac mass eq. 88.

Now we calculate the left-handed source eq. 54 projected onto $P^{+}$:

$\displaystyle\bar{S}^{+}_{LL}$ $\displaystyle=\frac{1}{2}{\rm
Tr}P^{+}\left([M,G^{>}]+[M,G^{<}]+[\Pi^{>},G^{h}]+[\Pi^{<},G^{h}]+\left(\\{\Pi^{>},G^{<}\\}-\\{\Pi^{<},G^{>}\\}\right)\right)_{LL}$
$\displaystyle=\frac{1}{2}{\rm
Tr}(P^{+})\left(m_{LR}(G_{RL,-}^{>}+G^{<}_{RL,-})-(G_{LR,+}^{>}+G^{<}_{LR,+})m_{RL}+2(\Pi^{>}_{LL}G^{<}_{LL,+}-\Pi^{<}_{LL,+}G^{>}_{LL})\right)$
$\displaystyle=0.$ (90)

Only the commutator with the tree-level Dirac mass $M_{0}$ and the collision
term contribute, and they add up to zero. There is thus no fermionic VIA
source at all orders in the vev-insertion expansion.

##  5 Conclusion

In this work we have shown that the resonantly enhanced source term
traditionally derived in the vev insertion approximation gets cancelled
exactly when all terms in the Kadanoff-Baym equations at leading order in the
derivative expansion are included. The derivation of the original VIA source
relied crucially on the existence of non-trivial flavor dynamics, i.e. a non-
alignment between the propagating and flavor eigenstates. In the case of
fermions, the term ‘flavor’ is (ab)used to refer to the two different
chiralities, which obtain distinct thermal corrections from the plasma. In our
analysis we have followed the original VIA literature, and allowed for flavor-
dependent interactions (chiral interactions for fermions) with the thermal
bath. This generalizes the results of Kainulainen:2021oqs , which considered
vector -like interactions.

We suspect that the cancellation of the contribitions to the source went
unnoticed in the original VIA literature, because of the use of 1-particle-
reducible (non-1PI) self-energies. Instead, our analysis relies solely on 1PI
self-energies, and uses a systematic perturbative expansion. In hindsight our
result may have been foreseen. Indeed, the VIA expansion source arises at
leading order in the derivative expansion, at which order no distinction
between a constant and an evolving background is made – and certainly no
source term is expected in the former case.

Interestingly, the problem with the original derivation of the VIA source term
does not stem from a pathology in the vev insertion expansion itself, or from
the existence of so-called pinch singularities (which are absent when the
fully resummed propagator solutions are used), but instead from the approach
based on non-IP1 diagrams that obscures a proper counting of all
contributions.

To show the cancellation of the source one must include the collision term in
the kinetic equation, which is non-zero because of the non-trivial flavor
dynamics. The assumption is that the interactions with the thermal bath are
fast, and to a good approximation the bath can be regarded as a collection of
particles in flavor eigenstates. The thermal self-energies then satisfy KMS
relations eq. 12, formulated in terms of the number densities of flavor
eigenstates. The spectral function, on the other hand, contains information on
the mass spectrum of the system. If the mass and flavor states are not
aligned, the Wightman functions no longer satisfy KMS relations in terms of
the number densities of flavor eigenstates, and thus the collision term does
not necessarily vanish. Our explicit expressions for the collision term, see
e.g. eq. 48, indeed vanish in the limit that the flavor off-diagonal mass term
$m_{LR}$ is set to zero, and flavor mixing is absent.

Computations of the baryon asymmetry using the VIA source suggested a rosier
picture of the feasibility of EWB in a number of BSM models. The larger yield
was mainly due to the CP-violating VIA source containing only one spacetime
derivative, and consequently there is no partial cancellation in particle
asymmetries when integrated over the bubble wall, in sharp contrast to the
semi-classical and flavor oscillation sources Cline:2021dkf . We do expect
that the VIA approximation gives a non-zero source term at the next (first)
order in the derivative expansion, and that the inclusion of thermal effects
may lead to resonant enhancement. But it remains to be seen whether this will
be enough to save large classes of models from the ever tightening constraints
imposed by the non-observation of EDMs.

## Acknowledgements

MP is supported by the Netherlands Organization for Scientific Research (NWO).
JvdV is supported by the Deutsche Forschungsgemeinschaft under Germany’s
Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306. GW is supported
by World Premier International Research Center Initiative (WPI Initiative),
MEXT, Japan. We thank Jordy de Vries and Thomas Konstandin for helpful
discussions. This work benefitted from discussions at the 2021 Lorentz Center
workshop ‘Computations that Matter’.

##  Appendix A Weyl basis

In this appendix we list all propagators and self-energies expressed in the
Weyl basis, defined as 2-by-2 subblocks of the matrices in Dirac space.

$\displaystyle\not{k}$
$\displaystyle=\left(\begin{array}[]{cc}0&k^{0}-\mathbf{k}\cdot\boldsymbol{\sigma}\\\
k^{0}+\mathbf{k}\cdot\boldsymbol{\sigma}&0\end{array}\right)=\left(\begin{array}[]{cc}0&K_{-}P^{+}+K_{+}P^{-}\\\
K_{+}P^{+}+K_{-}P^{-}&0\end{array}\right),$ (95) $\displaystyle\Pi^{h}$
$\displaystyle=\left(\begin{array}[]{cc}0&\\!a_{R}(k^{0}\\!-\\!\mathbf{k}\cdot\boldsymbol{\sigma})\\!+\\!b_{R}\\!\\\
\\!a_{L}(k^{0}\\!+\\!\mathbf{k}\cdot\boldsymbol{\sigma})\\!\\!+b_{L}\\!&0\end{array}\right)\\!=\\!\left(\begin{array}[]{cc}0&\\!\Pi^{h}_{RR,-}P^{+}\\!+\\!\Pi^{h}_{RR,+}P^{-}\\!\\\
\\!\Pi^{h}_{LL,+}P^{+}\\!+\\!\Pi^{h}_{LL,-}P^{-}\\!&0\end{array}\right),$
(100) $\displaystyle M_{0}$
$\displaystyle=\left(\begin{array}[]{cc}m_{RL}&0\\\
0&m_{LR}\end{array}\right),$ (103) $\displaystyle\Pi^{\lambda}$
$\displaystyle=\left(\begin{array}[]{cc}0&\Pi^{\lambda}_{RR}\\\
\Pi^{\lambda}_{LL}&0\end{array}\right)=\left(\begin{array}[]{cc}0&\bar{\gamma}_{R}g^{\lambda}_{R}\\\
\bar{\gamma}_{L}g^{\lambda}_{L}&0\end{array}\right),$ (108) $\displaystyle
G^{ab}$ $\displaystyle=\left(\begin{array}[]{cc}G_{LR}^{ab}&G^{ab}_{LL}\\\
G^{ab}_{RR}&G_{RL}^{ab}\end{array}\right)=\left(\begin{array}[]{cc}G^{ab}_{LR,+}P^{+}+G^{ab}_{LR-}P^{-}&\,G^{ab}_{LL,+}P^{+}+G^{ab}_{LL-}P^{-}\\\
G^{ab}_{RR,-}P^{+}+G^{ab}_{RR,+}P^{-}&\,G^{ab}_{RL,-}P^{+}+G^{ab}_{RL,+}P^{-}\end{array}\right),$
(113)

where we defined $K_{\pm}=k^{0}\pm k$.

##  Appendix B Particle and hole modes

In this appendix we give the connection between our propagator solutions to
the usual form of the Dirac propagators – with particle and hole modes – used
in the derivation of the VIA source in the literature, see in particular Ref.
Lee:2004we . In contrast to the analysis in Section 4, but in correspondence
with Weldon:1999th ; Lee:2004we , we will work with 4-dimensional Dirac
spinors in this appendix.

The self-energies for the left and right chiralities may differ in a chiral
theory and we split

$\Pi^{\alpha}=\Pi_{LL}^{\alpha}+\Pi_{RR}^{\alpha}=P_{R}\Pi^{\alpha}P_{L}+P_{L}\Pi^{\alpha}P_{R},$
(114)

with $\Pi^{\alpha}=\Pi^{h}\mp i\Pi^{\cal A}$. For a massless fermion the self-
energy can be written as

$\Pi_{II}^{h}=P_{J}\left(a_{I}\not{k}+b_{I}\not{u}\right)P_{I},\qquad\Pi_{II}^{\cal
A}=P_{J}\left(2\bar{\Gamma}_{I}\gamma_{0}\right)P_{I},$ (115)

with $J\neq I$. The solution to the leading order constraint equation for the
retarded/advanced Green’s function,
$(G^{r})^{-1}=(G_{LL}^{r})^{-1}+(G_{RR}^{r})^{-1}=P_{R}(G^{r})^{-1}P_{L}+P_{L}(G^{r})^{-1}P_{R}$,
is

$\displaystyle(G_{II}^{r})^{-1}$
$\displaystyle=P_{J}\left((A_{I,0}+2i\bar{\Gamma}_{I})\gamma^{0}-A_{I,s}\boldsymbol{\gamma}\cdot\hat{\mathbf{k}}\right)P_{I}=D_{I,+}\Lambda^{JI}_{+}+D_{I,-}\Lambda^{JI}_{-},$
(116)

with $I=L,R$. In the middle expression the coefficients are

$A_{I,0}=k_{0}(1-a_{I})-b_{I},\quad A_{I,s}=k(1-a_{I}),$ (117)

and on the RHS we rewrote this defining

$\displaystyle D_{I,\pm}=A_{I,0}+2i\bar{\Gamma}_{I}\mp
A_{I,s},\qquad\Lambda_{\pm}=\frac{1}{2}(\gamma^{0}\pm\boldsymbol{\gamma}\cdot\hat{\mathbf{k}}),\qquad\Lambda_{\pm}^{IJ}=P_{I}\Lambda_{\pm}P_{J}\;\;{\rm
for}\;\;I\neq J.$ (118)

The advanced propagator is of the same form with
$i\bar{\Gamma}\to-i\bar{\Gamma}$.

To invert the propagator we use that $\Lambda^{\pm}\Lambda^{\pm}=0$ and
$\Lambda^{+}\Lambda^{-}+\Lambda^{-}\Lambda^{+}=1$. Then
$G^{r}=G_{LL}^{r}+G_{RR}^{r}=P_{L}G^{r}P_{R}+P_{R}G^{r}P_{L}$ with

$\displaystyle
G_{II}^{r}=\frac{\Lambda^{IJ}_{-}}{D_{I,+}}+\frac{\Lambda^{IJ}_{+}}{D_{I,-}}.$
(119)

The poles of the propagators are located at $k_{0}=\mathcal{E}_{I,p}$ (and
$k_{0}=-\mathcal{E}^{*}_{I,p}$), which is the solution of $D_{I,+}=0$, and at
$k_{0}=\mathcal{E}_{I,h}$ (and $k_{0}=-\mathcal{E}^{*}_{I,h}$), which is the
solution of $D_{I,-}=0$. The real parts of the poles $E_{I}={\rm
Re}(\mathcal{E}_{I})$ correspond to the particle and hole energies (antiholes
and antiparticles) respectively, as indicated by the subscript. For the
retarded propagator the poles of the propagator have negative imaginary parts,
and in the narrow width approximation

$\mathcal{E}_{I,p}=E_{I,p}-i\Gamma_{I,p},\quad\mathcal{E}_{I,h}=E_{I,h}-i\Gamma_{I,h},$
(120)

with $\Gamma_{I}=Z_{I,p}\bar{\Gamma}_{I}$ and

$Z_{p,h}=(\partial_{k_{0}}D_{\pm})^{-1}|_{k_{0}=\mathcal{E}_{p,h}}=\frac{E^{2}_{p,h}-k^{2}}{2m_{T}^{2}},$
(121)

the residues of the poles Weldon:1999th ; Lee:2004we . The retarded propagator
can then be written as

$\displaystyle G_{II}^{r}=\Big{(}\frac{Z_{I,p}}{k_{0}-\mathcal{E}_{I,p}}$
$\displaystyle+\frac{Z_{I,h}^{*}}{k_{0}+\mathcal{E}_{I,h}^{*}}-f_{I}(k_{0},k)\Big{)}\Lambda_{-}^{IJ}$
$\displaystyle+\Big{(}\frac{Z_{I,h}}{k_{0}-\mathcal{E}_{I,h}}+\frac{Z_{I,p}^{*}}{k_{0}+\mathcal{E}_{I,p}^{*}}+f_{I}^{*}(-k_{0}^{*},k)\Big{)}\Lambda_{+}^{IJ}.$
(122)

The function $f_{I}$ gives the non-pole part of the propagator, which is
subdominant near a resonance. The poles at
$k_{0}=-\mathcal{E}^{*}_{I,p},-\mathcal{E}^{*}_{I,h}$ correspond to
excitations of anti-particles and anti-holes. The spectral function is
$\rho=\left(G^{r}-G^{a}\right)=\rho_{L}+\rho_{R}=P_{L}\rho P_{R}+P_{R}\rho
P_{L}$ with,

$\rho_{I}=P_{I}\left(G^{r}-G^{a}\right)P_{J}=\Lambda_{-}^{IJ}\rho_{I,+}+\Lambda_{+}^{IJ}\rho_{I,-},$
(123)

and

$\displaystyle\rho_{I,+}=\frac{Z_{I,p}}{k_{0}-\mathcal{E}_{I,p}}-\frac{Z_{I,p}^{*}}{k_{0}-\mathcal{E}_{I,p}^{*}}+\frac{Z_{I,h}^{*}}{k_{0}+\mathcal{E}_{I,h}^{*}}-\frac{Z_{I,h}}{k_{0}+\mathcal{E}_{I,h}}+...$
(124)

and $\rho_{I,-}(k_{0},k)=(\rho_{I,+}(-k_{0}^{*},k))^{*}$, and the elipses
denoting the sub-dominant non-pole contributions (see Bellac:2011kqa  for
their explicit form). The Wightman functions, generalized for non-zero
chemical potential in Ref. Lee:2004we  666Note that Ref. Lee:2004we  uses
$\bar{m}_{T}^{2}=2m_{T}^{2}$ and there is no factor $1/2$ in the residue.
Their dispersion relation (A.18) differs a factor 2 compared to Bellac:2011kqa
; Weldon:1982bn  given their definition of the thermal mass
$\bar{m}_{T}^{2}=g^{2}C_{2}(r)T^{2}/2$. , are

$\displaystyle
G_{II}^{\lambda}=g_{I}^{\lambda}\left(\Lambda_{-}^{IJ}\rho_{I,+}+\Lambda_{+}^{IJ}\rho_{I,-}\right),$
(125)

with $n_{I}(k^{0}-\mu_{I})=(e^{(k_{0}-\mu_{I})/T}+1)^{-1}$ the Fermi-Dirac
distribution.

In the hard thermal loop approximation

$\displaystyle A_{I,0}=k_{0}-\frac{m_{I}^{2}}{k_{0}}xQ_{0}(x),\quad
A_{I,s}=k+\frac{m_{I}^{2}}{k}(1-xQ_{0}(x)),$ (126)

with $Q_{0}=\frac{1}{2}\ln\frac{x+1}{x-1}$ the Legendre function of the 2nd
kind, $x=k_{0}/k$, and $m_{I}^{2}\equiv g^{2}C^{I}_{2}T^{2}/8$, with
$C_{2}^{I}$ denoting the quadratic Casimir operator of the gauge group under
which particle $I$ is charged. In the limit $k\to 0$ of zero momentum,
$xQ_{0}\to 1$, which gives $E_{I,p}=E_{I,h}=m_{I}$ and $Z_{I,p}=Z_{I,h}=1/2$.
This motivates the definition of the thermal mass $m_{I}$.

In the limit $k\to\infty$ the hole contribution becomes subdominant
$Z_{I,h}\ll Z_{I,p}$. Already for $k\gtrsim m_{I}$ this becomes a good
approximation and this limit was often used in the literature on the VIA
source, to avoid the complication of hole modes. The particle pole is located
at $E_{I,p}=k+m^{2}_{I}/k$, which is the high momentum limit of a particle
with energy $E_{I,p}=\sqrt{k^{2}+2m_{I}^{2}}$, and $Z_{p}=1$. In this case it
is more useful to define the thermal mass as $\bar{m}_{I}^{2}=2m_{I}^{2}$.

## References

  * (1) V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov, _On the Anomalous Electroweak Baryon Number Nonconservation in the Early Universe_ , _Phys. Lett. B_ 155 (1985) 36.
  * (2) M.E. Shaposhnikov, _Baryon Asymmetry of the Universe in Standard Electroweak Theory_ , _Nucl. Phys. B_ 287 (1987) 757.
  * (3) L.D. McLerran, M.E. Shaposhnikov, N. Turok and M.B. Voloshin, _Why the baryon asymmetry of the universe is approximately 10**-10_ , _Phys. Lett. B_ 256 (1991) 451.
  * (4) A.G. Cohen, D.B. Kaplan and A.E. Nelson, _Baryogenesis at the weak phase transition_ , _Nucl. Phys. B_ 349 (1991) 727.
  * (5) A.G. Cohen, D.B. Kaplan and A.E. Nelson, _Progress in electroweak baryogenesis_ , _Ann. Rev. Nucl. Part. Sci._ 43 (1993) 27 [hep-ph/9302210].
  * (6) M. Trodden, _Electroweak baryogenesis_ , _Rev. Mod. Phys._ 71 (1999) 1463 [hep-ph/9803479].
  * (7) J.M. Cline, _Baryogenesis_ , in _Les Houches Summer School - Session 86: Particle Physics and Cosmology: The Fabric of Spacetime_ , 9, 2006 [hep-ph/0609145].
  * (8) D.E. Morrissey and M.J. Ramsey-Musolf, _Electroweak baryogenesis_ , _New J. Phys._ 14 (2012) 125003 [1206.2942].
  * (9) G.A. White, _A Pedagogical Introduction to Electroweak Baryogenesis_ , _IOP concise series_ (2016) .
  * (10) B. Garbrecht, _Why is there more matter than antimatter? Calculational methods for leptogenesis and electroweak baryogenesis_ , _Prog. Part. Nucl. Phys._ 110 (2020) 103727 [1812.02651].
  * (11) A.D. Sakharov, _Violation of $CP$ invariance, $c$ asymmetry, and baryon asymmetry of the universe_, _Pis’ma Zh. Eksp. Teor. Fiz._ 5 (1967) 32.
  * (12) ACME collaboration, _Improved limit on the electric dipole moment of the electron_ , _Nature_ 562 (2018) 355.
  * (13) J.M. Cline, K. Kainulainen and M. Trott, _Electroweak Baryogenesis in Two Higgs Doublet Models and B meson anomalies_ , _JHEP_ 11 (2011) 089 [1107.3559].
  * (14) S. Liebler, S. Profumo and T. Stefaniak, _Light Stop Mass Limits from Higgs Rate Measurements in the MSSM: Is MSSM Electroweak Baryogenesis Still Alive After All?_ , _JHEP_ 04 (2016) 143 [1512.09172].
  * (15) G.C. Dorsch, S.J. Huber, T. Konstandin and J.M. No, _A Second Higgs Doublet in the Early Universe: Baryogenesis and Gravitational Waves_ , _JCAP_ 05 (2017) 052 [1611.05874].
  * (16) J.M. Cline, _Is electroweak baryogenesis dead?_ , 1704.08911.
  * (17) J. de Vries, M. Postma, J. van de Vis and G. White, _Electroweak Baryogenesis and the Standard Model Effective Field Theory_ , _JHEP_ 01 (2018) 089 [1710.04061].
  * (18) D. Bodeker and W. Buchmuller, _Baryogenesis from the weak scale to the grand unification scale_ , _Rev. Mod. Phys._ 93 (2021) 035004 [2009.07294].
  * (19) J.M. Cline and K. Kainulainen, _Electroweak baryogenesis at high bubble wall velocities_ , _Phys. Rev. D_ 101 (2020) 063525 [2001.00568].
  * (20) J.M. Cline and B. Laurent, _Electroweak baryogenesis from light fermion sources: A critical study_ , _Phys. Rev. D_ 104 (2021) 083507 [2108.04249].
  * (21) J. De Vries, M. Postma and J. van de Vis, _The role of leptons in electroweak baryogenesis_ , _JHEP_ 04 (2019) 024 [1811.11104].
  * (22) E. Fuchs, M. Losada, Y. Nir and Y. Viernik, _$CP$ violation from $\tau$, $t$ and $b$ dimension-6 Yukawa couplings - interplay of baryogenesis, EDM and Higgs physics_, _JHEP_ 05 (2020) 056 [2003.00099].
  * (23) M. Joyce, T. Prokopec and N. Turok, _Nonlocal electroweak baryogenesis. Part 2: The Classical regime_ , _Phys. Rev. D_ 53 (1996) 2958 [hep-ph/9410282].
  * (24) J.M. Cline, M. Joyce and K. Kainulainen, _Supersymmetric electroweak baryogenesis_ , _JHEP_ 07 (2000) 018 [hep-ph/0006119].
  * (25) K. Kainulainen, T. Prokopec, M.G. Schmidt and S. Weinstock, _First principle derivation of semiclassical force for electroweak baryogenesis_ , _JHEP_ 06 (2001) 031 [hep-ph/0105295].
  * (26) T. Prokopec, M.G. Schmidt and S. Weinstock, _Transport equations for chiral fermions to order h bar and electroweak baryogenesis. Part 1_ , _Annals Phys._ 314 (2004) 208 [hep-ph/0312110].
  * (27) T. Prokopec, M.G. Schmidt and S. Weinstock, _Transport equations for chiral fermions to order h-bar and electroweak baryogenesis. Part II_ , _Annals Phys._ 314 (2004) 267 [hep-ph/0406140].
  * (28) T. Konstandin, T. Prokopec and M.G. Schmidt, _Kinetic description of fermion flavor mixing and CP-violating sources for baryogenesis_ , _Nucl. Phys._ B716 (2005) 373 [hep-ph/0410135].
  * (29) T. Konstandin, T. Prokopec, M.G. Schmidt and M. Seco, _MSSM electroweak baryogenesis and flavor mixing in transport equations_ , _Nucl. Phys. B_ 738 (2006) 1 [hep-ph/0505103].
  * (30) V. Cirigliano, C. Lee and S. Tulin, _Resonant Flavor Oscillations in Electroweak Baryogenesis_ , _Phys. Rev. D_ 84 (2011) 056006 [1106.0747].
  * (31) A. Riotto, _Towards a nonequilibrium quantum field theory approach to electroweak baryogenesis_ , _Phys. Rev. D_ 53 (1996) 5834 [hep-ph/9510271].
  * (32) A. Riotto, _Supersymmetric electroweak baryogenesis, nonequilibrium field theory and quantum Boltzmann equations_ , _Nucl. Phys. B_ 518 (1998) 339 [hep-ph/9712221].
  * (33) M. Carena, J.M. Moreno, M. Quiros, M. Seco and C.E.M. Wagner, _Supersymmetric CP violating currents and electroweak baryogenesis_ , _Nucl. Phys. B_ 599 (2001) 158 [hep-ph/0011055].
  * (34) P. Huet and A.E. Nelson, _Electroweak baryogenesis in supersymmetric models_ , _Phys. Rev. D_ 53 (1996) 4578 [hep-ph/9506477].
  * (35) A. Riotto, _The More relaxed supersymmetric electroweak baryogenesis_ , _Phys. Rev._ D58 (1998) 095009 [hep-ph/9803357].
  * (36) M. Postma, _A different perspective on the vev insertion approximation for electroweak baryogenesis_ , _JHEP_ 09 (2021) 055 [2107.05971].
  * (37) K. Kainulainen, _CP-violating transport theory for electroweak baryogenesis with thermal corrections_ , _JCAP_ 11 (2021) 042 [2108.08336].
  * (38) P. Huet and E. Sather, _Electroweak baryogenesis and standard model CP violation_ , _Phys. Rev. D_ 51 (1995) 379 [hep-ph/9404302].
  * (39) P. Huet and A.E. Nelson, _CP violation and electroweak baryogenesis in extensions of the standard model_ , _Phys. Lett. B_ 355 (1995) 229 [hep-ph/9504427].
  * (40) C. Lee, V. Cirigliano and M.J. Ramsey-Musolf, _Resonant relaxation in electroweak baryogenesis_ , _Phys. Rev. D_ 71 (2005) 075010 [hep-ph/0412354].
  * (41) V. Cirigliano, C. Lee, M.J. Ramsey-Musolf and S. Tulin, _Flavored Quantum Boltzmann Equations_ , _Phys. Rev._ D81 (2010) 103503 [0912.3523].
  * (42) C. Greiner and S. Leupold, _Stochastic interpretation of Kadanoff-Baym equations and their relation to Langevin processes_ , _Annals Phys._ 270 (1998) 328 [hep-ph/9802312].
  * (43) P.A. Henning, M. Blasone, R. Fauser and P. Zhuang, _Thermalization of a quark - gluon plasma_ , hep-ph/9612492.
  * (44) M. Postma and J. van De Vis, _Source terms for electroweak baryogenesis in the vev-insertion approximation beyond leading order_ , _JHEP_ 02 (2020) 090 [1910.11794].
  * (45) E. Braaten and R.D. Pisarski, _Calculation of the quark damping rate in hot QCD_ , _Phys. Rev. D_ 46 (1992) 1829.
  * (46) H.A. Weldon, _Structure of the quark propagator at high temperature_ , _Phys. Rev. D_ 61 (2000) 036003 [hep-ph/9908204].
  * (47) M.L. Bellac, _Thermal Field Theory_ , Cambridge Monographs on Mathematical Physics, Cambridge University Press (3, 2011), 10.1017/CBO9780511721700.
  * (48) H.A. Weldon, _Effective Fermion Masses of Order gT in High Temperature Gauge Theories with Exact Chiral Invariance_ , _Phys. Rev. D_ 26 (1982) 2789.