# A different perspective on the vev insertion approximation for electroweak
baryogenesis

Marieke Postma<EMAIL_ADDRESS>Nikhef, Theory Group, Science Park 105, 1098
XG, Amsterdam, The Netherlands

###### Abstract

In the vev insertion approximation (VIA) the spacetime dependent part of the
mass matrix is treated as a perturbation. We calculate the source terms for
baryogenesis expanding both the self-energy and propagator to first order in
mass insertions, which gives the same results as the usual approach of
calculating the self-energy at second order and using zeroth order
propagators. This procedure shows explicitly the equivalence between including
the mass in the free or in the interaction Lagrangian. The VIA source then
originates from the same term in the kinetic equation as the semi-classical
source, but at leading order in the derivative expansion (the expansion in
diamond operators). On top, another type of derivative expansion is done,
which we estimate to be valid for a bubble width larger than the inverse
thermal width. This cuts off the divergence in the VIA source in the limit
that the thermal width vanishes.

††preprint:  Nikhef-2021-016

##  1 Introduction

In electroweak baryogenesis the matter-antimatter asymmetry of the universe is
created during the electroweak phase transition Bochkarev:1990fx ;
Cohen:1990py ; Cohen:1990it ; Turok:1990zg . The scenario relies on new
physics at the electroweak scale, which can be probed by the LHC
Arhrib:2013oia ; Chen:2013rba ; Chang:2017ynj ; CMS:2016tgd ; Englert:2014uua
; Brivio:2017vri ; Han:2009ra ; Boudjema:2015nda ; Ellis:2015dha ;
Askew:2015mda ; Demartin:2015uha , electric dipole moment measurements
ACME:2018yjb ; Chupp:2017rkp ; Balazs:2016yvi ; Chien:2015xha ;
Cirigliano:2016nyn ; deVries:2017ncy ; Andreev:2018ayy , and possibly future
gravitational wave observatories caprini2016science ; Caprini:2019egz ;
Crowder:2005nr . This requires accurate theoretical predictions.

The dynamics of the phase transition is a non-equilibrium, non-perturbative,
and finite-temperature process, and to arrive at transport equations
describing the (phase space) densities of the particles in the plasma various
approximations and expansions have to be made Garbrecht:2018mrp ; Joyce:1994zn
; Joyce:1994zt ; Cline:2000nw ; Kainulainen:2001cn ; Kainulainen:2002th ;
Prokopec:2003pj ; Konstandin:2013caa ; Konstandin:2004gy ; Prokopec:2004ic ;
Konstandin_2005 ; Fromme:2006wx ; Cline:2020jre ; Cirigliano:2011di ;
Cirigliano:2009yt ; Riotto:1995hh ; Riotto:1997vy ; Lee:2004we ; Carena:2000id
. Different theoretical approaches have been shown to give predictions for the
asymmetry that may vary by more than an order of magnitude Konstandin:2013caa
; Cline:2020jre .

In this paper we take a closer look at the vev insertion approximation (VIA),
in which the source term in the transport equations is derived expanding in
the spacetime dependent part of the mass matrix Riotto:1995hh ; Riotto:1997vy
; Lee:2004we . It is, however, not clear how exactly the sources in VIA are
related to the source terms derived in other approximation schemes. In the
standard derivation the VIA source derives from the collision term in the
kinetic equations, which is puzzling, as this contribution is absent in
approaches that do not expand in mass perturbations. Moreover, it has been
observed that the VIA source term diverges (in the degenerate mass limit) in
the limit that the thermal width is taken to zero, while it is not clear what
the physical origin of this enhancement is Cline:2020jre .

The VIA source can be derived from the Schwinger-Dyson equations, which in
turn can be reformulated in terms of the Kadanoff-Baym (KB) equations. The
spacetime dependent mass is treated as an interaction, resulting in non-zero
self energies. In the standard derivation the self-energy is calculated to
second order in VIA, that is, from diagrams with two insertions of the mass,
while the propagators are those of the free theory (dressed with thermal
corrections) Riotto:1995hh ; Riotto:1997vy ; Lee:2004we . Using these
expressions in the collision term in the kinetic equation then gives the
source term at second order in VIA. However, the self-energy diagram is not
one-particle irreducible (1PI) in that it can be split by cutting a single
line. In the usual resummation of self-energies in the Schwinger-Dyson
equations one would only include the 1PI diagrams, to avoid double counting.
At second order in VIA we still expect to get the correct results, but this
approach may obscure the origin of the source term.

Instead, we will calculate both the self-energy and the propagator at first
order in VIA to calculate the source term at 2nd order, and show that this
gives the same final result for the source term as the standard approach. For
simplicity we work with a scalar toy model, but we expect the results to be
straightforwardly generalizable to fermions as well. At first order the
(hermitian) self-energy is nothing but a mass insertion. This immediately
makes it clear that it is equivalent to (1) include the mass term $\delta
m^{2}(z)$ in the free Lagrangian, and then expand the KB equations in $\delta
m^{2}(z)$; or (2) include the mass term in the interaction Lagrangian, and
then expand the KB equations in first order self-energy $\Pi^{h}_{(1)}$. The
VIA source thus derives from the same term $\propto[-i\delta
m^{2}+\Pi^{h},G^{\lambda}]$, with $G^{\lambda}$ the Wightman function, in the
kinetic KB equation as the classical source term.

The VIA source is obtained at leading order in the derivative expansion, that
is, at leading order in an expansion of the diamond operator of the Wigner
space KB equation Riotto:1995hh ; Riotto:1997vy ; Lee:2004we . To evaluate the
resulting integral expression, on top another type of derivative expansion is
done, which assumes the mass insertion – and thus the bubble wall background –
varies slowly compared to the typical scales that dominate the integral. We
estimate this expansion to be valid for bubble wall widths
$L_{w}\gg\max\left(\Gamma_{T}^{-1},\bar{m}_{T}^{-1}\right)$, with $\Gamma_{T}$
thermal width and $\bar{m}_{T}^{2}$ the constant diagonal mass term including
thermal corrections. This cuts off the earlier mentioned divergence in the
source term for $\Gamma_{T}\to 0$, as in this limit the approximations used
are no longer valid.

This paper is organized as follows. In the next section 2 we briefly introduce
our scalar toy model. All results are formulated in general terms, and can
easily be adapted to other set-ups. Section 3 reviews the derivation of the KB
equations from the Schwinger-Dyson equations, and the transformation to Wigner
space. In section 4 we then focus on the derivation of the source in VIA.
Section 4.1 discusses the the self-energy at first order in VIA and the
equivalence between absorbing the mass perturbation in the free or in the
interaction Lagrangian. We calculate the source from the self-energy and
Green’s functions both expanded to first order in section 4.2, and show that
the results are equivalent to the usual approach in section 4.3. Finally, we
estimate the validity of the derivative-like expansion done to arrive at the
final VIA results in section 4.4. We end with concluding remarks.

##  2 Scalar toy model

For simplicity we will consider a scalar model in this paper, which avoids
complication with spin projections, but we expect the results can be rather
straightforwardly generalized to fermionic models as well.

Consider then a two-flavor scalar model with CP violating couplings to the
bubble background Cirigliano:2011di ; Cirigliano:2009yt . The quadratic
Lagrangian for the flavor doublet $\phi=(\phi_{L}\;\phi_{R})^{T}$ is

$\mathcal{L}=(\partial_{\mu}\phi)^{\dagger}(\partial^{\mu}\phi)-\phi^{\dagger}M^{2}(z)\phi$
(1)

with mass matrix

$M^{2}=M_{D}^{2}+\delta M^{2}(z)=\left(\begin{array}[]{cc}m_{L}^{2}&0\\\
0&m_{R}^{2}\end{array}\right)+\left(\begin{array}[]{cc}0&\delta m^{2}(z)\\\
(\delta m^{2}(z))^{*}&0\end{array}\right).$ (2)

The flavor-diagonal masses $m^{2}_{L,R}$ are constant, while the off-diagonal
mass $\delta m^{2}=|\delta m^{2}|{\rm e}^{i\theta}$ depends on the bounce
solution $v_{b}=v_{b}(z)$ describing the bubble wall. In VIA the off-diagonal
term is treated as an interaction, and the mass and flavor eigenstates
coincide for the free Lagrangian. In addition, $\phi$ interacts with the
degrees of freedom in the plasma. These plasma effects are incorporated
dressing the Green’s functions with a non-zero self-energy. Essential for the
VIA mechanism is that this self-energy correction is complex, and gives apart
from a thermal mass also a thermal width correction Riotto:1995hh ;
Riotto:1997vy .

For future reference we define the Klein-Gordon operator ${\cal D}(u)$ via

$\displaystyle S=-\int{\rm
d}^{4}z\,\phi(z)^{\dagger}\mathcal{D}(z)\phi(z)=-\int{\rm
d}^{4}z\,\phi(z)^{\dagger}(\partial_{z}^{2}+M^{2}(z))\phi(z).$ (3)

##  3 Kadanoff-Baym equations

In this section we briefly review the derivation of the Kadanoff-Baym (KB)
equations that govern the quantum dynamics of the system. This also serves to
set the notation.

###  3.1 CTP formalism and Green functions

The Closed Time Path (CTP) or Schwinger-Keldysh formalism can describe finite-
temperature and non-equilibrium quantum systems Schwinger:1960qe ;
Keldysh:1964ud ; Chou:1984es . The path integral is defined along a contour
${\cal C}$ that starts from an initial time $t_{0}\to-\infty$ to time
$t=\infty$ and then back, which can be split in a ${\cal C}_{+}$ and ${\cal
C}_{-}$ branch. The scalar Green function defined via

$G(u,v)=\langle\Omega|T_{\cal C}\,\phi(u)\phi^{\dagger}(v)|\Omega\rangle,$ (4)

with $T_{\cal C}$ denoting time-ordering along the contour ${\cal C}$, can be
split depending on which branch the time-arguments of the fields lie. We use
the notation that
$G^{t}=G^{++},\,G^{\bar{t}}=G^{--},\,G^{>}=G^{-+},\,G^{<}=G^{+-}$.

The retarded and advanced propagators, and hermitian and anti-hermitian
propagators are

$\displaystyle G^{r}$ $\displaystyle\equiv G^{t}-G^{<}=G^{>}-G^{\bar{t}},$
$\displaystyle G^{a}$ $\displaystyle\equiv G^{t}-G^{>}=G^{<}-G^{\bar{t}},$
$\displaystyle G_{h}$
$\displaystyle\equiv\frac{1}{2}(G^{r}+G^{a})=\frac{1}{2}(G^{t}-G^{\bar{t}}),$
$\displaystyle G_{ah}$
$\displaystyle\equiv\frac{1}{2i}(G^{a}-G^{r})=\frac{i}{2}(G^{>}-G^{<}).$ (5)

A further useful relation is

$G^{t}(u,v)+G^{\bar{t}}(u,v)=G^{>}(u,v)+G^{<}(u,v).$ (6)

The hermiticity properties are $G^{\lambda}(u,v)^{\dagger}=G^{\lambda}(v,u)$
with $\lambda=>,\,<$ and $G^{t}(u,v)^{\dagger}=G^{\bar{t}}(v,u)$.

Similarly, one can define the advanced/retarded and hermitian/anti-hermitian
scalar self-energies in terms of $\Pi^{ab}$, with $a,b=+,\,-$.

###  3.2 From Schwinger-Dyson to Kadanoff-Baym

The starting point are the Schwinger-Dyson equations, derived form the 2PI
effective action in Prokopec:2003pj ; Garbrecht:2018mrp , which can be written
in the form

$\displaystyle\mathcal{D}(u)G^{ab}(u,v)$
$\displaystyle=-ai\delta_{ab}\delta^{4}(u-v)-i\sum_{c}c\int
d^{4}w\,\Pi^{ac}(u,w)G^{cb}(w,v),$ $\displaystyle
G^{ab}(u,v)\stackrel{{\scriptstyle\leftarrow}}{{\mathcal{D}}}(v)$
$\displaystyle=-ia\delta_{ab}\delta^{4}(u-v)-i\sum_{c}c\int
d^{4}w\,G^{ac}(u,w)\Pi^{cb}(w,v),$ (7)

with ${\cal D}$ the Klein-Gordon operator eq. 3 and $a,b=+,\,-$. For
$\lambda=>,<$ they become

$\displaystyle\partial_{u}^{2}G_{uv}^{\lambda}$
$\displaystyle=-M^{2}_{u}G_{uv}^{\lambda}-i\int{\rm
d}^{4}w\,\left(\Pi^{h}_{uw}G^{\lambda}_{wv}+\Pi^{\lambda}_{uw}\
G^{h}_{wv}+\frac{1}{2}\left(\Pi^{>}_{xz}G^{<}_{zy}-\Pi^{<}_{xz}G^{>}_{zy}\right)\right),$
$\displaystyle\partial_{v}^{2}G_{uv}^{\lambda}$
$\displaystyle=-G_{uv}^{\lambda}M^{2}_{v}-i\int{\rm
d}^{4}w\,\left(G^{h}_{uw}\Pi^{\lambda}_{wv}+G^{\lambda}_{uw}\Pi^{h}_{wv}+\frac{1}{2}\left(G^{>}_{xz}\Pi^{<}_{zy}-G^{<}_{xz}\Pi^{>}_{zy}\right)\right),$
(8)

where we introduced the shorthand notation $A_{uw}=A(u,w)$. Now add and
subtract these equations to obtain the anti-hermitian and hermitian part,
which correspond to the constraint and kinetic equation respectively:

$\displaystyle(\partial^{2}_{u}+\partial^{2}_{v})G^{\lambda}_{uv}$
$\displaystyle=-\\{M^{2},G^{\lambda}\\}_{uv}-i\int{\rm
d}^{4}w\,\left(\\{\Pi^{h},G^{\lambda}\\}+\\{\Pi^{\lambda},G^{h}\\}+{\cal
C}^{-}_{\rm coll}\right)_{uv}$ (9)
$\displaystyle(\partial^{2}_{u}-\partial^{2}_{v})G^{\lambda}_{uv}$
$\displaystyle=-[M^{2},G^{\lambda}]_{uv}-i\int{\rm
d}^{4}w\,\left([\Pi^{h},G^{\lambda}]+[\Pi^{\lambda},G^{h}]+{\cal C}^{+}_{\rm
coll}\right)_{uv}$ (10)

with the shorthand notation
$\left[\Pi,G\right]_{uv}=\Pi_{uw}G_{wu}-G_{uw}\Pi_{wu}$ and similar for the
anti-commutators. The collision term is

$({\cal C}^{\pm}_{\rm
coll})_{uv}=\frac{1}{2}\left([\Pi^{>},G^{<}]^{\pm}-[\Pi^{<},G^{>}]^{\pm}\right)_{uv}$
(11)

with $[..,..]^{+}=\\{..,..\\}$ anti-commutator and $[..,..]^{-}=[..,..]$
commutator. Here we used eq. 5 to rewrite the right hand side

$\displaystyle\Pi^{>}G^{t}-\Pi^{\bar{t}}G^{>}$
$\displaystyle=\Pi_{h}G^{>}+\Pi^{>}G_{h}+\frac{1}{2}\left(\Pi^{>}G^{<}-\Pi^{<}G^{>}\right),$
$\displaystyle\Pi^{t}G^{<}-\Pi^{<}G^{\bar{t}}$
$\displaystyle=\Pi_{h}G^{<}+\Pi^{<}G_{h}+\frac{1}{2}\left(\Pi^{>}G^{<}-\Pi^{<}G^{>}\right).$
(12)

Equations 9 and 10 are the position space Kadanoff-Baym (KB) equations.

###  3.3 Wigner representation

To do a gradient expansion in the slowly varying bubble background, it is
useful to define the relative and center-of-mass coordinates

$x=\frac{1}{2}(u+v),\quad r=u-v\qquad\Leftrightarrow\qquad
u=x+\frac{1}{2}r,\quad v=x-\frac{1}{2}r.$ (13)

We use the notation for a general function $A(u,v)$

$A(u,v)=A(x+\frac{1}{2}r,x-\frac{1}{2}r)\equiv\bar{A}(r,x).$ (14)

The Wigner transform is defined as the Fourier transform with respect to the
relative coordinate

$A(k,x)=\int{\rm d}^{4}(u-v)\,{\rm e}^{ik.(u-v)}A(u,v)=\int{\rm d}^{4}r\,{\rm
e}^{ik.r}\bar{A}(r,x).$ (15)

Now integrate eq. 8 with a factor $\int{\rm d}^{4}(u-v){\rm e}^{ik.(u-v)}$,
and add and subtract the two equations to extract the hermitian and anti-
hermitian parts

$\displaystyle\left(\frac{1}{2}\partial_{x}^{2}-2k^{2}\right)G^{\lambda}(k,x)$
$\displaystyle=-i{\rm
e}^{-i\diamond}\left(\\{-iM^{2}+\Pi^{h},G^{\lambda}\\}+\\{\Pi^{\lambda},G^{h}\\}+{\cal
C}^{-}_{\rm coll}\right),$ (16) $\displaystyle
2ik\cdot\partial_{x}G^{\lambda}(k,x)$ $\displaystyle=i{\rm
e}^{-i\diamond}\left([-iM^{2}+\Pi^{h},G^{\lambda}]+[\Pi^{\lambda},\
G^{h}]+{\cal C}^{+}_{\rm coll}\right).$ (17)

where all $G^{i},\,\Pi^{i}$ are a function of $(k,x)$, and $M^{2}$ a function
of $(x)$. The collision term eq. 11 in Wigner space is

${\cal C}^{\pm}_{\rm
coll}(k,x)=\frac{1}{2}\left([\Pi^{>}(k,x),G^{<}(k,x)]^{\pm}-[\Pi^{<}(k,x),G^{>}(k,x)]^{\pm}\right).$
(18)

The diamond operator is defined as

$\diamond\big{(}A(k,x)B(k,x)\big{)}=\frac{1}{2}\big{(}\partial_{x}A(k,x)\cdot\partial_{k}B(k,x)-\partial_{k}A(k,x)\cdot\partial_{x}B(k,x)\big{)}.$
(19)

Some useful relations involving the diamond operator, used in the above
derivation, are collected in appendix A.

Equations 16 and 17 are the Wigner space Kadanoff-Baym equations,
corresponding to the constraint and kinetic equation respectively.

##  4 VEV insertion approximation

In the vev insertion approximation (VIA) the off diagonal and space-time
dependent parts of the mass matrix, $\delta M^{2}$ in eq. 2, are treated as a
perturbation. Flavor oscillations and coherence effects are neglected. The
source terms are calculated expanding the KB equations in the number of mass
insertions, which should converge for sufficiently small enough mass
perturbations $\delta m^{2}/(\bar{m}_{T}\Gamma_{T})\ll 1$, with
$\bar{m}_{T,i}^{2}$ the flavor diagonal mass including thermal corrections and
$\Gamma_{T}$ the thermal width Postma:2019scv .

In the usual derivation of the source term in VIA the self-energy is
calculated for two mass insertions, while the Green’s functions of the free
theory (dressed with thermal corrections) are used Riotto:1995hh ;
Riotto:1997vy ; Lee:2004we . As we will show in section section 4.3 this gives
the same result as calculating the both the self-energy and the Green’s
function to first order in VIA. We first follow the latter approach.

###  4.1 Self-energy at first order in VIA

It is clear that we can either (1) include $\delta M^{2}$ in the free
Lagrangian, derive the KB equations, and expand in $\delta M$, or (2) include
$\delta M^{2}$ in the interaction Lagrangian which result in a non-zero self-
energy, derive the KB equations and expand in powers of the self-energy. These
are two different descriptions of the same expansion of the same physical
system. We proceed with the latter approach and set $\delta M^{2}=0$ in the KB
equations, and incorporate the mass insertions via the self-energy.

At first order in VIA (one mass insertion) the non-zero self energies in
position space are

$\Pi_{ij,(1)}^{ab}(u,v)=f_{ij}^{ab}(u)\delta^{4}(u-v),$ (20)

with

$\displaystyle f_{LR}^{++}(u)$ $\displaystyle=-i\delta m^{2}(u),$
$\displaystyle f_{LR}^{--}(u,)$ $\displaystyle=i\delta m^{2}(u),$
$\displaystyle f_{RL}^{++}(u)$ $\displaystyle=-i(\delta m^{2}(u))^{*},$
$\displaystyle f_{RL}^{--}(u,v)$ $\displaystyle=i(\delta m^{2}(u))^{*}.$ (21)

The vertices on the ${\cal C}^{-}$ branch pick up an additional minus sign.
Throughout we will use $a,b,..=+,-$ for CTP contour indices and $i,j,..=L,R$
for flavor indices. The subscript $(\\#)$ on $\Pi^{ab}_{ij}$ and $G^{ab}_{ij}$
refers to the number $\\#$ of mass insertions. Flavor diagonal self-energies
vanish at this order, as well as $\Pi^{\lambda}$, as vertices always are
inserted on the same branch of the CTP contour. We note that the hermitian
self-energy

$\Pi_{LR,(1)}^{h}(u,v)=\frac{1}{2}\left(\Pi_{LR,(1)}^{++}(u,v)-\Pi_{LR,(1)}^{--}(u,v)\right)=-i\delta
m^{2}(v)\,\delta^{4}(u-v)$ (22)

is proportional to the mass perturbation, whereas the orthogonal combination
$\propto(\Pi_{LR,(1)}^{++}+\Pi_{LR,(1)}^{--})=0$ vanishes. If we insert these
results in the position space KB eqs. 9 and 10, the integral over $\Pi_{h}$
can be done with the delta-function, and using eq. 21, the result is

$\displaystyle(\partial^{2}_{u}+\partial^{2}_{v})G^{\lambda}_{uv}$
$\displaystyle=-\\{M_{D}^{2}+\delta M^{2},G^{\lambda}\\}_{uv},$
$\displaystyle(\partial^{2}_{u}-\partial^{2}_{v})G^{\lambda}_{uv}$
$\displaystyle=[M_{D}^{2}+\delta M^{2},G^{\lambda}]_{uv}.$ (23)

This indeed gives exactly the same equations as one would have obtained by
absorbing $\delta M^{2}$ in the free Lagrangian and setting $\Pi_{h}=0$.

From the KB equations in Wigner space eqs. 16 and 17 it is even more
transparant that both procedures give the same result, as these only depend on
the combination $(-i\delta M^{2}+\Pi^{h})$. The Wigner space self-energy is

$\Pi_{ij,(1)}^{ab}(k,x)=\int{\rm d}^{4}r\,{\rm
e}^{ikr}f^{ab}_{ij}(x+\frac{1}{2}r)\delta(r)=f^{ab}_{ij}(x),$ (24)

and thus $\Pi_{LR,(1)}^{h}=-i\delta m^{2}(x)$, which depends on the collective
coordinate $x=u=v$, but not on the momenta conjugate to the relative
coordinates.

The above implies that the source term in VIA can be equally derived from an
expansion in $(-i\delta M^{2})$ as from an expansion in $\Pi_{h}$. The source
term thus should arise from

$2ik\cdot\partial_{x}G^{\lambda}(k,x)=i{\rm
e}^{-i\diamond}[-i(M_{D}^{2}(x)+\delta M^{2}(x))+\Pi^{h}(x),G^{\lambda}(k,x)]$
(25)

in the kinetic equation. This is the same term that gives rise to the semi-
classical force Joyce:1994zt ; Joyce:1994fu ; Huber:2000ih ; Cline:1997vk ;
Cline:2000nw . The KB equations resums all mass insertions; instead in VIA the
kinetic equation is solved perturbatively expanding in mass insertions.

###  4.2 Source term from first order self-energy

To derive the transport equations at 2nd order in the VIA expansion, we
perform a derivative expansion of the kinetic equations in Wigner space, and
then transform the results back to position space. The propagators are most
easily derived directly in position space from the Schwinger-Dyson equations.
We will calculate the source at 2nd order expanding both the self-energy and
Green’s functions to first order in VIA. The self-energy was discussed in the
previous section.

Setting $\delta M^{2}=0$ in the free Lagrangian, the mass matrix is flavor
diagonal, and the zeroth order in VIA constraint equation for the Green’s
function becomes

$\displaystyle(\partial_{u}^{2}+m_{i}^{2})G^{ab}_{ij,(0)}(u-v)=-ia\delta_{ab}\delta^{4}(u-v)$
(26)

for $i=L,R$ flavors. The mass matrix is diagonal and the solutions are the
usual free thermal propagators Landsman:1986uw . Coherence effects are
neglected ($f_{LR}$ distributions are set to zero) and the off-diagonal
propagators vanish. In VIA the propagators are dressed with a non-zero self-
energy to account for the interaction with the thermal plasma Parwani:1991gq ,
which amounts to adopting the spectral function Riotto:1995hh

$\rho(k)=i\left[\frac{1}{(k^{0}+i\epsilon+i\Gamma_{T})^{2}-\omega^{2}}-\frac{1}{(k^{0}-i\epsilon-i\Gamma_{T})^{2}-\omega^{2}}\right],$
(27)

with $\omega^{2}=\vec{k}^{2}+m_{i}^{2}+m_{T,i}^{2}$ and
$m^{2}_{T},\,\Gamma_{T}$ the thermal mass and width respectively Riotto:1995hh
. For vanishing plasma corrections $m^{2}_{T},\,\Gamma_{T}\to 0$ this reduces
to the free spectral function.

At first order in VIA the Schwinger-Dyson equation reads

$\displaystyle G^{ab}_{(1),ij}(u,v)$ $\displaystyle=\sum_{cd}cd\int{\rm
d}^{4}z_{1}\int{\rm
d}^{4}z_{2}\,G^{ac}_{(0),ik}(u-z_{1})\Pi^{cd}_{(1),kl}(z_{1}-z_{2})G^{db}_{(0),lj}(z_{2}-v).$
(28)

Using the non-zero first order self-energy eq. 21 and zeroth order Green’s
function this gives

$\displaystyle G_{(1),ij}^{++}(u,v)$ $\displaystyle=\sum_{cd}\int{\rm
d}^{4}z_{1}\int{\rm
d}^{4}z_{2}\,G^{+-}_{(0),ii}(u-z_{1})\Pi^{--}_{(1),ij}(z_{1}-z_{2})G^{-+}_{(0),jj}(z_{2}-v)$
$\displaystyle=\int{\rm
d}^{4}z\,G^{+-}_{(0),LL}(u-z)f^{--}_{ij}(z)G^{-+}_{(0),RR}(z-v),$
$\displaystyle G_{(1),ij}^{--}(u,v)$ $\displaystyle=\int{\rm
d}^{4}z\,G^{-+}_{(0),ii}(u-z)f^{++}_{ij}(z)G^{+-}_{(0),jj}(z-v),$ (29)

for $i\neq j$, and as before $i,j=L,R$.

The source term can be derived from the kinetic equation eq. 17, which at 2nd
order becomes

$\displaystyle k\cdot\partial_{x}\left(G^{>}+G^{<}\right)_{ij}$
$\displaystyle=\frac{1}{2}{\rm
e}^{-i\diamond}\left([\Pi^{h},G^{t}+G^{\bar{t}}]+[\Pi_{i}^{t}+\Pi_{i}^{\bar{t}},\
G^{h}]+{\cal C}^{+}_{\rm coll}\right)_{ij}$
$\displaystyle\stackrel{{\scriptstyle(2)}}{{=}}\frac{1}{2}{\rm
e}^{-i\diamond}\left([\Pi_{(1)}^{h},G_{(1)}^{t}+G_{(1)}^{\bar{t}}]\right)\delta_{ij}.$
(30)

On the first line we used eq. 6, and on the 2nd line we gave the 2nd order VIA
contribution originating from multiplying a 1st order 1PI self-energy with a
1st order propagator. Integrating over momenta gives the derivative of the
number current (particles minus antiparticles) density for the left hand side.
Using eq. 24 to write $\Pi_{(1)}^{h}(k,x)=f^{h}(x)$, which only depends on the
collective coordinate $x$, this gives

$\displaystyle\partial_{\mu}j_{(2),i}^{\mu}(x)$
$\displaystyle\equiv\int\frac{{\rm
d}^{4}k}{(2\pi)^{4}}\,k^{\mu}\,\left(G_{(2)}^{<}(k,x)+G_{(2)}^{>}(k,x)\right)_{ii}$
$\displaystyle=\frac{1}{2}\int\frac{{\rm d}^{4}k}{(2\pi)^{4}}\,{\rm
e}^{-i\diamond}\left([f^{h}(x),G_{(1)}^{t}(k,x)+G_{(1)}^{\bar{t}}(k,x)]\right)_{ii}.$
(31)

The right hand side is known as the source term $S$, i.e., we write
$\partial_{\mu}j_{(2),i}^{\mu}(x)=S_{(2),i}$ at second order in VIA. The first
two terms in the derivative expansion, corresponding to an expansion of ${\rm
e}^{-i\diamond}=1-i\diamond+{\cal O}(\diamond^{2})$, are

$\displaystyle S_{(2),i}(x)|_{\textsc{LO}}$
$\displaystyle=\frac{1}{2}\int\frac{{\rm
d}^{4}k}{(2\pi)^{4}}\,[f^{h}(x),G_{(1)}^{t}(k,x)+G_{(1)}^{\bar{t}}(k,x)]_{ii},$
$\displaystyle S_{(2),i}(x)|_{\textsc{NLO}}$
$\displaystyle=-\frac{i}{2}\\{\partial_{x}f^{h}(x),\partial_{k}(G_{(1)}^{t}(k,x)+G_{(1)}^{\bar{t}}(k,x))\\}_{ii},$
(32)

with LO and NLO denoting the leading and next-to-leading order in the
derivative expansion. To transform back the results to coordinate space we use
that

$\displaystyle\int\frac{{\rm d}^{4}k}{(2\pi)^{4}}\,f^{h}(x)G(k,x)$
$\displaystyle=\int\frac{{\rm d}^{4}k}{(2\pi)^{4}}\int{\rm d}^{4}r\,{\rm
e}^{ikr}f^{h}(x)\bar{G}(r,x)=f^{h}(x)\bar{G}(0,x)=f^{h}(u)G(u,u),$
$\displaystyle\int\frac{{\rm
d}^{4}k}{(2\pi)^{4}}\,\partial_{x}f^{h}(x)\partial_{k}G(k,x)$
$\displaystyle=\int\frac{{\rm d}^{4}k}{(2\pi)^{4}}\int{\rm
d}^{4}r\,(ir)\partial_{x}f^{h}(x)\bar{G}(r,x){\rm e}^{ikr}=0.$ (33)

Thus only the leading order term gives a non-zero result; all higher order
terms in the derivative expansion vanish as they are proportional to $r^{n}$
(and the $k$-integral gives a $\delta(r)$). We thus only get a source term at
leading order in the derivative expansion
$S_{(2),ii}(u)=S_{(2),ii}\big{|}_{\textsc{LO}}$ with

$\displaystyle S_{(2),ii}$
$\displaystyle=\frac{1}{2}\left[f^{h}(u),G_{(1)}^{t}(u,u)+G_{(1)}^{\bar{t}}(u,u)\right]_{ii}$
$\displaystyle={\rm Re}\int{\rm
d}^{4}w\Big{(}f^{\bar{t}}_{ij}(u)G^{<}_{(0),jj}(u,w)f^{t}_{ji}(w)G^{>}_{(0),ii}(w,u)$
$\displaystyle\hskip
59.75095pt-f^{t}_{ij}(u)G^{>}_{(0),jj}(u,w)f^{\bar{t}}_{ji}(w)G^{<}_{(0),ii}(w,u)\Big{)}.$
(34)

On the 2nd line we rewrote the first order Green’s function in terms of the
lowest order Green’s functions using eq. 29. This is the same result as in the
standard calculation, as shown in the next subsection.

###  4.3 Source term from 2nd order self-energy

The standard approach the source is derived using the 0th order Green’s
functions and the 2nd order self-energy. In this subsection we briefly review
this derivation, and show that the resulting source term is the same as
derived in the previous subsection.

The kinetic equation eq. 10 for $G^{>}+G^{<}=G^{t}+G^{\bar{t}}$ can be written
in the form

$\displaystyle(\partial^{2}_{u}\\!-\\!\partial^{2}_{v})(G^{t}\\!+\\!G^{\bar{t}})_{uv}$
$\displaystyle=-i\\!\int\\!{\rm
d}^{4}w\left([\Pi^{++},G^{++}]-[\Pi^{--},G^{--}]+\\{\Pi^{-+},G^{+-}\\}-\\{\Pi^{+-},G^{-+}\\}\right)_{uv}$
(35)

where for notational convenience we suppressed flavor indices. The last two
terms on the right hand side originate from the collision term. At second
order this becomes

$\displaystyle(\partial^{2}_{u}-\partial^{2}_{v})(G^{t}+G^{\bar{t}})_{uv}$
$\displaystyle\stackrel{{\scriptstyle(2)}}{{=}}-i\int{\rm
d}^{4}w\,\left([\Pi_{(1)}^{++},G_{(1)}^{++}]-[\Pi_{(1)}^{--},G_{(1)}^{--}]\right)_{uv}$
$\displaystyle\stackrel{{\scriptstyle(2)}}{{=}}-i\int{\rm
d}^{4}w\,\left(-\\{\Pi_{(2)}^{+-},G_{(0)}^{-+}\\}+\\{\Pi_{(2)}^{-+},G_{(0)}^{+-}\\}\right)_{uv},$
(36)

where the first line is the approach of the previous subsection, and the 2nd
line the standard derivation. Including both would be double counting. To show
that both expressions are indeed the same, we start with the self-energy at
2nd order in VIA

$\Pi_{(2),ii}^{ab}(u,v)=cd\int{\rm d}^{4}z_{1}\int{\rm
d}^{4}z_{2}\,\Pi_{(1),ij}^{ac}(u,z_{1})G_{(0),jj}^{cd}(z_{1},z_{2})\Pi_{(1),ji}^{db}(z_{2},v).$
(37)

$+$$+$$-$$-$$\underbrace{-\Pi^{++}_{(1),ij}\odot
G_{(0),jj}^{+-}\odot\Pi^{--}_{(1),ji}}_{\Pi^{+-}_{(2),ii}}$$\odot$$-$$+$$G^{-+}_{(0),ii}$$=$$-$$+$$+$$\Pi^{++}_{(1),ij}$$\odot$$-$$-$$+$$-$$\underbrace{G_{(0),jj}^{+-}\odot\Pi^{--}_{(1),ji}\odot
G_{(0),ii}^{-+}}_{G_{(1),ji}^{++}}$ Figure 1: The equivalence eq. 38 between
$\Pi^{-+}_{(2),ii}\odot G^{+-}_{(0),ii}=-\Pi^{--}_{(1),ij}\odot
G^{--}_{(1),ji}$, with $\odot$ denoting integration over the internal
variable, in terms of Feynman diagrams. A mass insertion is represented by a
cross, ${\times}$, in the diagrams.

It follows that

$\displaystyle\int{\rm d}^{4}w\,\\{\Pi_{(2)}^{+-}(u,w),G_{(0)}^{-+}(w,v)\\}$
$\displaystyle=-\int{\rm d}^{4}w\int{\rm d}^{4}z_{1}\int{\rm
d}^{4}z_{2}\,\Big{(}\Pi_{(1)}^{++}(u,z_{1})G_{(0)}^{+-}(z_{1},z_{2})\Pi_{(1)}^{--}(z_{2},w)G_{(0)}^{-+}(w,v)$
$\displaystyle\hskip
113.81102pt+G_{(0)}^{-+}(u,w)\Pi_{(1)}^{++}(w,z_{1})G_{(0)}^{+-}(z_{1},z_{2})\Pi_{(1)}^{--}(z_{2},v)\Big{)}$
$\displaystyle=-\int{\rm
d}^{4}z_{1}\,\left(\Pi_{(1)}^{++}(u,z_{1})G_{(1)}^{++}(z_{1},v)+G_{(1)}^{++}(z_{1},v)\Pi_{(1)}^{++}(u,z_{1})\right),$
(38)

where the expression for the 1st order propagator in eq. 29 was used. The
overall minus sign of $\Pi_{(2)}^{+-}$ arises from the definition eq. 37, as
the sum over the internal indices $c,d=+,-$ gives an extra sign. This
expression can be equally understood in terms of Feynman diagrams as shown in
Figure 1, where we also reinstated flavor indices. Likewise,

$\displaystyle\int{\rm
d}^{4}w\,\\{\Pi_{(2)}^{-+}(u,w),G_{(0)}^{+-}(w,v)\\}=-\\!\int{\rm
d}^{4}z_{1}\\!\left(\Pi_{(1)}^{--}(u,z_{1})G_{(1)}^{--}(z_{1},v)+G_{(1)}^{--}(z_{1},v)\Pi_{(1)}^{--}(u,z_{1})\right),$
(39)

which together proof that the two expressions in eq. 36 above are the same.

Let’s briefly review the derivation of the source in terms of the 2nd order
self-energy. The non-zero self-energies are

$\displaystyle\Pi_{(2),ii}^{-+}=-f_{ij}^{--}(u)G_{(0),jj}^{-+}(u,v)f_{ji}^{++}(v),\quad\Pi_{(2),ii}^{+-}$
$\displaystyle=-f_{ij}^{++}(u)G_{(0),jj}^{+-}(u,v)f_{ji}^{--}(v),$ (40)

which can be substituted in the kinetic equation eq. 36. For the left hand
side we use that111In Riotto:1995hh ; Riotto:1997vy ; Lee:2004we  the current
conservation is derived directly in position space in the limit $v\to u$ via
$\lim_{u\to
v}i\left(\partial^{2}_{u}-\partial^{2}_{v}\right)\left(G^{t}(u,v)+G^{\bar{t}}(u,v)\right)=2\partial_{\mu}j^{\mu}(x)$.

$\displaystyle\frac{i}{2}\left(\partial^{2}_{u}-\partial^{2}_{v}\right)\left(G^{t}(u,v)+G^{\bar{t}}(u,v)\right)$
$\displaystyle=\partial_{x^{\mu}}\int\frac{{\rm
d}^{4}k}{(2\pi)^{4}}\,k^{\mu}{\rm
e}^{-ikr}\left(G^{<}(k,x)+G^{>}(k,x)\right)=\partial_{\mu}j^{\mu}(x)$ (41)

with $x=\frac{1}{2}(u+v)$. Taking the leading order result in the diamond
expansion, corresponding to setting $v=u$ in position space, then gives the
source
$\partial_{\mu}j_{(2)}^{\mu}(x)|_{\textsc{LO}}=S_{(2),ii}|_{\textsc{LO}}$ with

$\displaystyle S_{(2),ii}|_{\textsc{LO}}$ $\displaystyle=\frac{1}{2}\int{\rm
d}^{4}w\,\left(-\\{\Pi_{(2)}^{+-},G^{-+}_{(0)}\\}+\\{\Pi^{-+}_{(2)},G^{+-}_{(0)}\\}\right)_{uu}=\int{\rm
d}^{4}w\,{\rm
Re}\left(\Pi^{-+}_{(2)}G_{(0)}^{+-}-\Pi_{(2)}^{+-}G^{-+}_{(0)}\right)_{uu}$
$\displaystyle={\rm Re}\int{\rm
d}^{4}w\Big{(}f^{\bar{t}}_{ij}(u)G^{<}_{(0),jj}(u,w)f^{t}_{ji}(w)G^{>}_{(0),ii}(w,u)$
$\displaystyle\hskip
51.21504pt-f^{t}_{ij}(u)G^{>}_{(0),jj}(u,w)f^{\bar{t}}_{ji}(w)G^{<}_{(0),ii}(w,u)\Big{)}.$
(42)

This agrees with eq. 34.

###  4.4 CP even and odd sources

The integral in the source eq. 42 cannot be evaluated analytically for a
generic background $f(x)$. To proceed it is assumed that the background is
slowly varying, which in addition allows to identify to dominant contribution
to the CP even and CP odd parts of the source. This expansion is not the usual
derivative expansion, in the sense of an expansion in powers of the diamond
operator in the Wigner space KB equations eq. 17; indeed eq. 42 is the leading
order term, and – as follows from eq. 33 – the only contribution from the
diamond expansion. What is done instead is that the mass insertion $f(w)$ in
the source eq. 42, with $w$ the coordinate that is integrated over, is
expanded as222At next-to-leading order, i.e. for the term with one derivative,
only the time component $\lambda=0$ yields a non-zero result.

$f(w)=f(u)+\partial_{\lambda}f(u)(w-u)^{\lambda}+{\cal O}(\partial^{2}).$ (43)

The $u$-dependent functions can then be taken out of the integral, and the
resulting integral becomes manageable. The leading and next-to-leading terms
in this expansion give a CP even and CP odd contribution to the source. For
the left flavor

$\displaystyle S_{L}^{\textsc{CP}}$ $\displaystyle=2|\delta m^{2}|^{2}\int{\rm
d}^{4}w\,{\rm
Re}\Big{(}G^{<}_{(0),RR}(u,w)G^{>}_{(0),LL}(w,u)-G^{>}_{(0),RR}(u,w)G^{<}_{(0),LL}(w,u)\Big{)}$
(44) $\displaystyle S_{L}^{\textsc{CPV}}$ $\displaystyle=2{\rm Im}\left[\delta
m^{2}(\delta m^{2})^{*^{\prime}}\right]\times$ $\displaystyle\hskip
28.45274pt\int{\rm d}^{4}w\,{\rm
Im}\left[(w-u)\left(G^{<}_{(0),RR}(u,w)G^{>}_{(0),LL}(w,u)-G^{>}_{(0),RR}(u,w)G^{<}_{(0),LL}(w,u)\right)\right],$

where we used eq. 21 for $f(u)$. The source for the right flavor is
$S_{R}=-S_{L}$.

The expansion eq. 43 is only valid if the background varies on much larger
length scales then the length scales dominating the integral in the
expressions above, otherwise the full spacetime dependence of the mass should
be taken into account. The mass insertion is a function of the background
bounce solution. For large bubbles curvature effects can be neglected, and the
bubble is generically well approximated by a kink solution, which in the
bubble wall rest frame takes the form John:1998ip

$v_{b}(u)\approx\frac{v_{N}}{2}\left(1+\tanh(\frac{u^{0}-u^{3}}{L_{w}})\right).$
(45)

Here $v_{N}$ is the vev in the bubble interior, $L_{w}$ the bubble wall width,
$u^{0}=v_{w}t$ with $v_{w}$ the bubble wall speed, and $u^{3}$ the distance
from the center of the wall.

The background thus varies on length scales of the bubble wall width. The
typical scales dominating the integral can be estimated from the explicit
expressions for the source terms; here we look at the CP-odd source which has
been calculated to give Riotto:1995hh ; Riotto:1997vy ; Lee:2004we

$\displaystyle S_{L}^{\rm CPV}$ $\displaystyle\propto\int\frac{k^{2}{\rm
d}k}{\omega_{L}\omega_{R}}{\rm
Im}\bigg{[}\frac{\left(n_{B}(\mathcal{E}_{L}^{*})-n_{B}(\mathcal{E}_{R})\right)}{(\mathcal{E}_{L}^{*}-\mathcal{E}_{R})^{2}}+\frac{\left(n_{B}(\mathcal{E}_{L})+n_{B}(\mathcal{E}_{R})\right)}{(\mathcal{E}_{L}+\mathcal{E}_{R})^{2}}\bigg{]}$
$\displaystyle\propto\int\frac{k^{2}{\rm d}k}{\omega^{2}}{\rm
Re}\left[\frac{{\rm e}^{\omega/T}}{\Gamma_{T}(1+{\rm e}^{\omega/T})^{2}}+{\cal
O}(\Gamma_{T})\right]_{\bar{m}_{T,L}^{2}=\bar{m}_{T,R}^{2}},$ (46)

with $n_{B}(\mathcal{E})$ is the Bose-Einstein distribution, and $k=|\vec{k}|$
the modulus of the three-momenta. The complex energies are defined via
$\mathcal{E}_{j}=\omega_{j}-i\Gamma_{j}$ with
$\omega_{j}=\sqrt{k^{2}+\bar{m}_{T,j}^{2}}$ with
$\bar{m}_{T,j}^{2}=m_{j}^{2}+m_{T,j}^{2}$ the sum of the diagonal zero-
temperature mass and the thermal mass, and $\Gamma_{T,j}$ the thermal decay
width. On the 2nd line in eq. 46 we took the limit of degenerate masses
$\bar{m}_{T,L}^{2}=\bar{m}_{T,R}^{2}\equiv\bar{m}_{T}^{2}$. This expression
shows that the integral over three momentum is cutoff by the mass term in
$\omega$ and the integral varies over $|\vec{k}|\sim\bar{m}_{T}$,
corresponding to a length scale $\bar{m}_{T}^{-1}$.

To find the typical $k^{0}$ momenta, and the typical scale of the conjugate
$u^{0}$ variation, it is useful to see how the denominators in the first line
of eq. 46 arise, namely from integrals of the form

$\int_{0}^{\infty}{\rm d}u^{0}\,u^{0}{\rm
e}^{ik^{0}u^{0}}=-\frac{1}{k_{0}^{2}},\qquad
k_{0}=\mathcal{E}_{L}^{*}-\mathcal{E}_{R},\,\mathcal{E}_{L}+\mathcal{E}_{R},$
(47)

with the specific $k_{0}$ values determined from contour integrals and the
poles of the propagators. The $u^{0}$-integral is cutoff by the thermal width
${\rm Im}(k_{0})\propto\Gamma_{T}$.

We thus conclude that the background changes slowly compared to the length
scales dominating the intergral, and can be safely approximated as nearly
constant, if333Refs. Riotto:1995hh ; Riotto:1997vy  cite the milder bound
$L_{w}\gg v_{w}\Gamma^{-1}_{T}$.

$L_{w}\gg\max\left[\Gamma^{-1}_{T},\bar{m}_{T}^{-1}\right],$ (48)

where the two terms on the right hand side come from the constraints on the
temporal and spatial variation respectively. If eq. 48 is violated the
expansion eq. 43 breaks down, and the resulting expressions for the source eq.
46 cannot be trusted.

The source term eq. 46 diverges for degenerate flavor-diagonal masses in the
limit $\Gamma_{T}\to 0$. This has been noted before, and questions on the
physical origin of this divergence were raised Cline:2020jre . We see now that
the expansion eq. 43 used to derive this result breaks down when the inverse
thermal width becomes larger than the bubble wall width
$\Gamma_{T}^{-1}>L_{w}$. This cuts off the divergence for any finite bubble
size.

Although for simplicity we have concentrated on the scalar model, the
derivation in VIA of the source for fermionic systems is very similar, and the
resulting source terms have a very similar structure to eq. 46. In particular,
the same expansion eq. 43 is made, and the same bound on the validity eq. 48
applies. We can thus apply this limit on the use of the VIA source also for
fermionic systems. As an explicit example, consider that the CP violation
originates from modified Yukawa interactions

${\cal L}_{\rm int}=-\delta m(z)\bar{\psi}_{L}\psi_{R}-\delta
m(z)^{*}\bar{\psi}_{R}\psi_{L},$ (49)

with $\psi_{i}$ the left- and right-hand chiralities of a Standard Model (SM)
fermion.444The left and right chiralities are the equivalent of the left and
right flavors of the scalar model, with the mass insertion off-diagonal in
left-right ‘flavor’ space.  In the Standard Model $\delta m$ is just the mass
term arising from the Yukawa coupling to the Higgs, but to obtain sufficient
CP violation requires new physics corrections to this, which we leave
unspecified. Including thermal corrections, crucial for the VIA source term,
the fermion propagator is dressed with a flavor diagonal thermal mass and
thermal width $m^{2}_{T,i}$ and $\Gamma_{T,i}$ with $i=L,R$. The thermal
corrections are of the order $m^{2}_{T}\sim\alpha T^{2}$ and
$\Gamma_{T}\sim\alpha T$, with $\alpha\sim\alpha_{3}$ the QCD coupling for
quarks and $\alpha=\alpha_{2}$ the electroweak coupling for leptons. The VIA
results are then valid, see eq. 48, for

$L_{w}T\gg\alpha^{-1}\sim\left\\{\begin{array}[]{rl}10,&\qquad{\rm for}\;{\rm
quarks},\\\ 30,&\qquad{\rm for}\;{\rm leptons},\end{array}\right.$ (50)

which requirers thick bubble walls.

##  5 Discussion

In this paper we have calculated the source term at second order in the vev
insertion approximation (VIA) from the self-energy and Green’s functions both
calculated at first order. Our approach agrees with the known results, but has
the advantage that it shows clearly the equivalence of expanding the transport
equations in terms of mass perturbations or in self-energies. The source term
then derives, as expected from this equivalence, from the same term in the
kinetic Kadanoff-Baym equation eq. 25 as semiclassical source terms based on a
gradient expansion. In particular, both the semiclassical WKB force and flavor
mixing forces can be derived from this term, see Garbrecht:2018mrp  and
references therein.

Despite their common origin the VIA and semiclassical source terms have
different parameteric dependence, and it thus appears that they are distinct
sources Garbrecht:2018mrp . The VIA source crucially depends on the CP-even
phase provided by the thermal width in the dressed propagator eq. 27, and in
fact vanishes in the limit $\Gamma_{T}\to 0$ for non-degenerate flavor
diagonal masses $m_{L}^{2}\neq m_{R}^{2}$, while the semiclassical source
terms survive this limit. It is clear that resumming the mass insertions in
VIA cannot yield the semiclassical force, and vice versa, the semi-classical
force can not be derived from an expansion in mass perturbations of the VIA
source. Their difference derives thus not so much from the expansion in mass
perturbations, but rather can be traced back to taking different moments of
the Kadanoff-Baym equation and a different derivative expansion.

In the semiclassical scheme the Wigner space KB-equations are integrated over
$k^{0}$-momenta, to obtain Boltzmann equations for the phase space densities.
The source arises at first order in the gradient expansion, i.e., in the
expansion in diamond operators.555For bosons, the WKB source at first order in
the gradient expansion is CP conserving, and a CP violating source can only
originate at higher order Konstandin_2005 ; Cline:2000nw . Instead, in the VIA
scheme the KB-equations are integrated over four-momenta, which projects out
all higher order terms in the the diamond expansion eq. 33, and thus also gets
rid of the semiclassical source. The resulting equations are formulated in
terms of number currents (and using the diffusion Ansatz, number densities)
rather than phase space densities.

To evaluate the integral in the VIA source term eq. 42 another derivative-type
expansion is performed eq. 43, which assumes the background varies slowly
compared to the length scales dominating the source integral. We estimate that
this expansion is valid, and the VIA results can be trusted, for bubble widths
$L_{w}\gg\max\left[\Gamma^{-1}_{T},\bar{m}_{T}^{-1}\right]$, with
$\bar{m}^{2}_{T},$ the flavor diagonal mass including thermal corrections, and
$\Gamma_{T}$ the thermal width. The thermal corrections arise from the plasma
interactions.

The VIA source has a resonance for equal mass terms for the two flavors in the
free Lagrangian, as can be seen from the explicit expression eq. 46. This
suggests it arises from (resonant) flavor oscillation dynamics. We would then
expect the resonance to shift by order $\delta m^{2}$-corrections if the mass
insertions are resummed and the difference between flavor and mass eigenstates
fully accounted for. As the VIA expansion only converges for small enough mass
perturbations, these corrections are relatively small. The VIA source was
calculated with four mass insertions in Postma:2019scv ; for degenerate flavor
diagonal masses this next-to-leading order term is small if

${\rm bosons}:\;\;\delta m^{4}\ll\bar{m}_{T}^{2}\Gamma_{T}^{2},\qquad{\rm
fermions}:\;\;\delta
m^{2}\ll\frac{\bar{m}_{T}^{2}\Gamma_{T}^{2}}{(\bar{m}_{T}^{2}+\Gamma_{T}^{2})}$
(51)

for bosons and fermions respectively Postma:2019scv .666For top quarks the vev
insertion expansion breaks down as eq. 51 is not satisfied, and the vev
insertions must be resummed. The resonance is cut off by the thermal width.

It will be interesting to further explore the relation between the VIA source
and semiclassical (resonant) flavor sources derived in an approach based on a
gradient expansion Cirigliano:2011di ; Cirigliano:2009yt . Although the
semiclassical approach can be derived withouth thermal corrections, the
divergence in the degenerate mass limit is unshielded (see e.g. eq. (174) in
Garbrecht:2018mrp ), and thermal corrections should be included for
theoretical consistency. To further assess and improve the estimate on the
validity of the VIA results, it will also be useful to explicitly calculate
the next term in the derivative expansion eq. 43. In addition, one could try
to devise alternative methods to evaluate the source term for small bubble
wall widths, to bypass the expansion completely. This is left for future work.

## Acknowledgments

The author thanks Jorinde van de Vis and Jordy de Vries for very useful
discussions. This work was funded by the Netherlands Organization for
Scientific Research (NWO).

##  Appendix A Diamond operator

To rewrite the KB equations in Wigner space we use the following identities
for the diamond operator

$\displaystyle\int{\rm d}^{4}(u-v){\rm e}^{ik.(u-v)}\int{\rm
d}^{4}wA(u,w)B(w,v)$ $\displaystyle={\rm
e}^{-i\diamond}\left(A(k,x)B(k,x)\right)$ $\displaystyle\int{\rm
d}^{4}(u-v){\rm e}^{ik.(u-v)}A(u)B(u,v)$ $\displaystyle={\rm
e}^{-i\diamond}\left(A(x)B(k,x)\right)$ $\displaystyle\int{\rm d}^{4}(u-v){\rm
e}^{ik(u-v)}\left(\partial^{A}_{u}+2\partial^{B}_{u}\right)A(u)B(u,v)$
$\displaystyle={\rm
e}^{-i\diamond}\left(\partial^{B}_{x}-2ik\right)A(x)B(k,x)$
$\displaystyle\int{\rm d}^{4}(u-v){\rm
e}^{ik(u-v)}\left(\partial^{A}_{v}+2\partial^{B}_{v}\right)B(u,v)A(v)$
$\displaystyle={\rm
e}^{-i\diamond}\left(\partial^{B}_{x}+2ik\right)A(x)B(k,x)$ (52)

To derive the first identity above, we rewrite the left hand side in terms of
the collective coordinate $x=\frac{1}{2}(u+v)$ and relative coordinate
$r=u-v=r_{1}+r_{2}$ with $r_{1}=u-w$, and $r_{2}=w-v$.

$\displaystyle A(u,w)$ $\displaystyle=\bar{A}(r_{1},x+\frac{1}{2}r_{2})={\rm
e}^{\frac{1}{2}r_{2}\partial_{x}^{A}}\bar{A}(r_{1},x)$ $\displaystyle B(w,v)$
$\displaystyle=\bar{B}(r_{2},x-\frac{1}{2}r_{1})={\rm
e}^{-\frac{1}{2}r_{1}\partial_{x}^{B}}\bar{B}(r_{2},x)$ (53)

where in the last step we used that $\partial_{x}$ is the generator of
translation, and the superscript denotes the term the derivative is acting on.
Then

$\displaystyle\int{\rm d}^{4}(u-v){\rm e}^{ik(u-v)}\int{\rm
d}^{4}w\,A(u,w)B(w,v)$ $\displaystyle\hskip 56.9055pt=\int{\rm
d}^{4}r_{1}\,{\rm
e}^{(ik-\frac{1}{2}\partial_{x}^{B})r_{1}}\bar{A}(r_{1},x)\int{\rm
d}^{4}r_{2}\,{\rm e}^{(ik+\frac{1}{2}\partial_{x}^{A})r_{2}}\bar{B}(r_{2},x)$
$\displaystyle\hskip
56.9055pt=A(k+\frac{i}{2}\partial_{x}^{B},x)B(k-\frac{i}{2}\partial_{x}^{A},x)={\rm
e}^{-i\frac{1}{2}\left(\partial_{x}^{A}\partial_{k}^{B}-\partial_{x}^{B}\partial_{k}^{A}\right)}A(k,x)B(k,x)$
$\displaystyle\hskip 56.9055pt={\rm e}^{-i\diamond}A(k,x)B(k,x).$ (54)

where in the penultimate step we used that $\partial_{k}$ is the generator of
translations in momentum space. The derivation of the other identities in eq.
52 can be done in a similar way.

The derivative expansion ${\rm e}^{-i\diamond}=1-i\diamond+...$. transformed
back to position space becomes

$\displaystyle\int\frac{{\rm d}^{4}k}{(2\pi)^{4}}{\rm
e}^{-i\diamond}A(k,x)B(k,x)$ $\displaystyle=\int\frac{{\rm
d}^{4}k}{(2\pi)^{4}}\int{\rm d}^{4}r\int{\rm d}^{4}r^{\prime}{\rm
e}^{-\frac{i}{2}\left(\partial_{x}^{A}\cdot\partial_{k}^{B}-\partial_{k}^{A}\cdot\partial_{x}^{B}\right)}{\rm
e}^{ik(r+r^{\prime})}\bar{A}(r,x)\bar{B}(r^{\prime},x)$
$\displaystyle=\int{\rm d}^{4}r\,{\rm
e}^{r\cdot\left(\partial_{x}^{A}+\partial_{x}^{B}\right)}\bar{A}(r,x)\bar{B}(-r,x)$
$\displaystyle=\int{\rm
d}^{4}w\left[1+(u-w)\cdot\left(\partial_{u}+\partial_{w}\right)+{\cal
O}(\partial^{2})\right]A(u,w)B(w,u).$ (55)

Note that the leading order derivative expansion corresponds to the limit
$v\to u$ in position space

$\displaystyle\int{\rm d}^{4}wA(u,w)B(w,v)\Big{|}_{\textsc{LO}}$
$\displaystyle=\int\frac{{\rm d}^{4}k}{(2\pi)^{4}}{\rm
e}^{-i\diamond}A(k,x)B(k,x)\Big{|}_{\textsc{LO}}=\int{\rm d}^{4}A(u,w)B(w,u).$
(56)

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