# AN EFFECTIVE FIELD THEORY STUDY OF MEDIUM HEAVY QUARK EVOLUTION

Miguel Ángel Escobedo<EMAIL_ADDRESS>Departament de Física
Quàntica i Astrofísica and Institut de Ciències del Cosmos, Universitat de
Barcelona, Martí i Franquès 1, 08028 Barcelona, Catalonia, Spain.

###### Abstract

The evolution of hard probes in a medium is a complex multiscale problem that
significantly benefits from the use of Effective Field Theories (EFTs). Within
the EFT framework, we aim to define a series of EFTs in a way that addresses
each energy scale individually in separate steps. However, studying hard
probes in a medium presents challenges. This is because an EFT is typically
constructed by formulating the most general Lagrangian compatible with the
problem’s symmetries. Nevertheless, medium effects may not always be encoded
adequately in an effective action. In this paper, we construct an EFT that is
valid for studying the evolution of a heavy quark in a QCD plasma, with the
temperature scale integrated out. Through this example, we explicitly
demonstrate how to handle the doubling of degrees that arise in non-
equilibrium field theory. As a result, we derive a Fokker-Planck equation
using only symmetry and power counting arguments. The methods introduced in
this paper will pave the way for future developments in the study of
quarkonium suppression.

##  I Introduction

The quark-gluon plasma is a state of matter that forms at high temperatures
and densities, in which quarks and gluons are not confined within hadrons.
This state of matter was present in the early universe and can also be created
on Earth in experiments using ultrarelativistic heavy-ion collisions. Such
experiments are currently conducted at the Large Hadron Collider at CERN in
Switzerland and at the Relativistic Heavy Ion Collider at Brookhaven National
Lab in the US. However, the quark-gluon plasma exists only for a very short
time during these collisions, so we must study the particles produced during
this brief period to learn about the properties of the plasma. One approach is
to study “hard probes,” which are observables that are both significantly
affected by the medium and can be measured in the challenging environment of a
heavy-ion collision. Examples of hard probes include heavy quarks, heavy
quarkonium, and jets.

Studying heavy particles and jets in a medium requires dealing with largely
separated energy scales. Particles with energies of the order of the heavy
quark mass or a hard parton energy are rare in the medium and can be
accurately described using perturbative QCD. However, particles with energies
of the order of the temperature are sensitive to the medium and cannot always
be described using perturbation theory. Therefore, it is interesting to
separate the contribution from these two kind of particles to encode non-
perturbative effects in parameters or functions that can be computed using
non-perturbative methods like lattice QCD. It is also important to be careful
when describing systems with largely separated energy scales. On one hand, the
appearance of largely separated energy scales can lead to a breaking of naive
perturbation theory, meaning that the size of a contribution cannot be
directly related to its number of loops. On the other hand, non-perturbative
studies using lattice QCD are also challenging because a very large lattice is
required to accommodate all the energy scales.

These problems can be solved by using EFTs. An EFT is a quantum field theory
that gives the same results as another more general theory at low energies.
They are constructed in the following way [1]:

  * •

Identify the relevant degrees of freedom and the symmetries of the problem.

  * •

Write the more general Lagrangian that respects the symmetries of the problem
using the relevant degrees of freedom.

  * •

An EFT must be equipped with a power counting. This means that there is a
simple rule to predict how large is the contribution of each term in the
Lagrangian for a given observable.

  * •

The unknown parameters in the effective Lagrangian are called Wilson
coefficients. They are fixed by imposing that the EFT gives the same results
as the full theory at low energies. This procedure is called matching.

Note that an EFT Lagrangian has an infinite number of terms. However, the
theory still has predictive power thanks to the power counting.

To illustrate how the use of EFTs can solve the problems discussed earlier,
let us consider a simple example. Imagine a theory consisting of a very heavy
particle with mass $M$ and some other massless particles. We can create an EFT
for energies much smaller than $M$. By construction, the mass $M$ only appears
giving the correct dimensions to the Wilson coefficients of the EFT.
Therefore, since the only active energy scale in the EFT is the momentum of
the particles which we call $p$, the size of a given term in the Lagrangian
can be determined by looking at the powers of $1/M$ that appear in the
corresponding Wilson coefficient. Perturbative expansions are then reorganized
as a double expansion in the coupling constant and $p/M$, in which the order
of magnitude of each contribution can be determined before performing the
computation. In the case of a lattice computation, since the high energy
effects are encoded in the Wilson coefficients, the lattice spacing can be
larger than $1/M$. This way, the needed lattice is much smaller than the one
that would be required for an equivalent computation using the full theory.

The use of EFTs can also allow relating physical observables with quantities
computable on the lattice in a more direct way. A good example is the
description of heavy quarkonium. Heavy quarkonium is a bound state made of a
heavy quark and a heavy antiquark. Quarkonium is a non-relativistic system in
which well separated energy scales appear. They are the mass, $M$, the inverse
of the typical radius, $1/r\sim Mv$ with $v\ll 1$, and the binding energy
$E\sim Mv^{2}$. Non-relativistic QCD (NRQCD) [2, 3] is an EFT valid for
energies smaller than $M$. Since $M\gg\Lambda_{QCD}$ we can reliably match QCD
to NRQCD using perturbation theory. We can also use Potential NRQCD (pNRQCD)
[4, 5, 6], an EFT valid for energies much smaller than $1/r$. Since this scale
is not always perturbative, there are cases in which the matching between
NRQCD and pNRQCD cannot be done in perturbation theory. However, since each
term in the pNRQCD Lagrangian has a scaling with $1/M$ that is easy to
determine, we can perform the matching between NRQCD and pNRQCD in the limit
in which the mass of the heavy quarks is infinite (the static limit). Once the
matching is done, we can use these Wilson coefficients to compute the
properties of quarkonium states with finite heavy quark masses. This is a
rigorous way to show that the spectrum of quarkonium can be determined by
solving a Schrödinger equation in which the potential is computed on the
lattice using static quarks.

It is natural to expect that EFTs are useful to study hard probes in a medium.
Such problems almost always imply the appearance of widely separated energy
scales, those related with the hard probe and the ones induced by the medium.
Generally speaking, perturbative and lattice QCD computations are more
challenging at finite temperature. However, using EFTs we can separate the
contribution from high energy scales (perturbative and unaffected by the
medium) from that of low energy scales (potentially non-perturbative and
sensitive to the medium). In this way, it often becomes possible to encode all
the information needed from the medium in transport coefficients or potentials
that might be computed using non-perturbative techniques. EFTs have been
applied to the study of hard probes in heavy ion collisions. In the case of
heavy quarkonium, medium modified versions of NRQCD and pNRQCD has been used
to compute the thermal corrections to the mass and the medium induced decay
width [7, 8, 9]. pNRQCD has also been used to study the evolution of the
population of bound states inside of a medium in the cases in which $1/r\gg T$
[10, 11, 12, 13]. Regarding jets, there are recently developed EFTs for the
study of jet broadening in [14] and jet sub-structure in heavy-ion collisions
[15].

However, the application of the EFT framework to the study of hard probes in a
medium presents a conceptual challenge. The procedure that we have outlined
before needs to be modified when the medium affects the matching between the
full theory and the EFT. At finite temperature, computations of non-static
properties are done using the so-called real-time formalism [16]. In this
formalism, there is a doubling of degrees of freedom. The reason is that the
state of the system is not determined by a wave function but instead by a
density matrix. Then, we need to introduce two different type of fields.
Fields of type 1 (2) that originate from the time-evolution acting on the left
(right) of the density matrix. In this way, correlators in which only fields
of type 1 (2) appear are chronologically (anti-chronologically) ordered.
However, in a given computation, we might need to evaluate correlators that
mix the two types of fields and that represent more complex time-orderings. In
these conditions, the matching between the full theory and the EFT might
induce terms including the two types of fields. Physically, we might interpret
this in the following way. The EFT describes the evolution of an open quantum
system interacting with an environment (the medium degrees of freedom that
have been integrated out). Then, the effect of high energy fields on low
energy ones can not be encoded in an effective action. We need a more general
object called influence functional instead [17]. This is a substantial
modification of the second point of the procedure we outlined to construct and
EFT. Up to now, this difficulty has been skipped in the study of hard probes
using two different strategies. The first one is to focus on computing the
binding energy and the decay width of non-relativisty systems. In this case,
the effects of the doubling of degrees of freedom are neutralized in the
approximation in which the heavy particles are dilute [7, 8]. The second
strategy is to study cases in which the temperature is smaller than the
energies that are integrated out to define the EFT [10, 11].

The construction of EFTs that do not consist in an effective action has been
investigated in the context of the development of an EFT for hydrodynamics
[18, 19, 20, 21]. However, to our knowledge, these developments have not been
applied to the study of hard probes in a QCD plasma. In this manuscript, we
aim to fill this gap by studying a simple example. A heavy quark close to
thermalization in a medium. In this problem, we can identify three widely
separated energy scales.

  * •

The mass of the heavy quark $M$.

  * •

The spatial momentum of the heavy quark $p$, which is of order of $\sqrt{TM}$,
being $T$ the temperature.

  * •

The temperature $T$ and other energy scales that the medium might induced.

It is well-known that this situation can be studied using a Fokker-Planck
equation or the physically equivalent Langevin equation [22, 23, 24, 25, 26,
27, 28]. In this manuscript, we will give an EFT perspective on the problem.
Following the EFT philosophy, we will deal with each energy scale in a
separate way. We will follow these steps:

  1. 1.

As a starting point, we can use NRQCD. This automatically encodes the effects
of degrees of freedom with energy of order $M$.

  2. 2.

It is straight-forward to define an EFT for energies smaller than $\sqrt{MT}$.
This EFT is a modification of NRQCD using the momentum-label technique common
in Soft-Collinear Effective Theory (SCET) [29, 30] and On-Shell Effective
Theory (OSEFT) [31, 32]. An equivalent EFT, with the same degrees of freedom,
symmetries and power counting, was previously introduced to study a completely
different problem. This is NRQCD for semi-hard fields ($\textrm{NRQCD}_{sh}$)
[33]. This EFT was originally introduced to study modes with energy of the
order $\sqrt{M\Lambda_{QCD}}$ in the study of heavy quarkonium at $T=0$. In
our case, the role of $\Lambda_{QCD}$ is played by $T$. However, symmetries
and power counting arguments remain the same.

  3. 3.

We define an EFT for energies below $T$. In this EFT there are terms that mix
fields of type $1$ and type $2$. These are dealt with following the approach
of [18] but supplementing it with the properties of the dilute expansion. We
call this EFT Langevin Effective Theory (LET) due to its connection with a the
Langevin equation.

  4. 4.

We perform the matching between $\textrm{NRQCD}_{sh}$ and LET in the one-gluon
exchange approximation as an illustration. This allows to encode the influence
of degrees of freedom with energy of order $T$.

  5. 5.

We use LET to compute the evolution of the density of heavy quarks, obtaining
a Fokker-Planck equation for the density that coincides with the evolution
resulting from a Langevin equation.

This procedure could seem extremely complex to obtain results that are already
well-known. However, our aim is to develop techniques that could be useful for
more interesting cases. In particular, we have in mind the study of the
evolution of the reduced density matrix of quarkonium in the regime
$T\sim\frac{1}{r}$. The problem that we study in this manuscript is simpler
than quarkonium in the regime $T\sim\frac{1}{r}$. However, in both cases we
are dealing with dilute non-relativistic particles. One important observation
that the framework developed here makes manifest is that the physics at the
scale $T$ can be studied in the static limit. In other words, the matching
between $\textrm{NRQCD}_{sh}$ and LET can be done in the static limit, and we
can use those Wilson coefficients to study the finite $M$ case. A similar
thing happens regarding the matching between NRQCD and pNRQCD at $T=0$. These
two results combined strongly support the idea that the matching between NRQCD
and pNRQCD at finite temperature can be also done in the static limit. We note
that the evolution of a static heavy quark-antiquark pair in terms of gauge-
invariant expectation values has already been discussed in [34].

As a complement to the previous study, we will also discuss the case in which
$p\sim T$. In this case, we integrate out the scales $p$ and $T$ at the same
time. In other words, we go directly from NRQCD to LET. We observe that the
structure of LET in this case is still the same as in the case $p\gg T$.
However, the Wilson coefficients are different. We observe that there is a
smooth transition between the cases $p\sim T$ and $p\gg T$, as it should.
Physically, if we start with a heavy quark at rest, it will start to gain
momentum due to broadening. As the momentum becomes larger, two complementary
things happen. On one hand, the Fokker-Planck equation becomes accurate. On
the other hand, the drag force becomes a leading order effect.

The present work has some similarities with [35], however the perspective is
quite different. In [35] the AdS/CFT correspondence in Supersymmetric Yang-
Mills is taken as starting point, while in this work we tried to be as
agnostic as possible about the properties of the medium.

The manuscript is organized as follows. In section II, we will discuss
$\textrm{NRQCD}_{sh}$. Next, in section III , we will present LET. Section IV
discusses the computation of the evolution of the distribution of heavy quarks
within LET. In section V, we discuss the matching between
$\textrm{NRQCD}_{sh}$ and LET. In the section VI, we consider the case of a
heavy quark with momentum $p\sim T$. Finally, in section VII, we give our
conclusions.

##  II $\textrm{NRQCD}_{sh}$

Our starting point is the NRQCD Lagrangian. To fix the notation, we write its
heavy quark sector

$\mathcal{L}_{\psi}=\psi^{\dagger}(x)\\{iD_{0}+\frac{\mathbf{D}^{2}}{2M}+c_{4}\frac{\mathbf{D}^{4}}{8M^{3}}+c_{F}g\frac{\boldsymbol{\sigma}\cdot\mathbf{B}}{2M}\\}\psi(x)+\cdots$
(1)

Here $D_{\mu}=\partial_{\mu}-igA_{\mu}$, $\sigma$ is a Pauli matrix and $E$
and $B$ are the chromoelectric and chromomagnetic fields. We use the following
power counting. The spatial momentum of the heavy quark scales like
$\sqrt{MT}$ plus a possible residual momentum of order $T$. The gauge field is
only sensitive to the scale $T$. In eq. (1) we have only written terms of
order $\psi^{\dagger}\psi\frac{T^{2}}{M}$ or lower. These are the terms that
we need in order to compute $\frac{1}{f}\frac{df}{dt}$ to order
$\frac{T^{2}}{M}$ where $f(p,\mathbf{R})$ is the distribution function of
heavy quarks. However, only the first two terms in the Lagrangian will end up
contributing. We define $f$ in terms of the ¡ propagator of heavy quarks. More
details will be given later.

Using NRQCD to study this problem is not completely optimal. As we mentioned
before, the spatial momentum of heavy quarks is of order $\sqrt{MT}$ while
each interaction with the medium changes the momentum of the heavy quark by an
amount of order $T$. This implies that the term
$\psi^{\dagger}\frac{\mathbf{D}^{2}}{2M}\psi$ hides contributions of different
sizes. Then the power counting is not completely clear. Each time that we
apply this term it is not obvious whether we will get a contribution of order
$T$, $T\sqrt{\frac{T}{M}}$ or $\frac{T^{2}}{M}$. At best, we can put an upper
bound on the size of the contribution. To improve the situation we can
introduce momentum-label fields as it is done in SCET and OSEFT. Let us divide
the spatial momentum of heavy quarks in two pieces

$\mathbf{p}+\mathbf{k}\,,$ (2)

where $\mathbf{p}$ is of order $\sqrt{MT}$ and $\mathbf{k}$ is a residual
momentum of order $T$. To improve this we can perform the following
transformation

$\psi(t,\mathbf{x})=\sum_{\mathbf{p}}\mathrm{e}^{i\mathbf{p}\mathbf{x}}\xi_{\mathbf{p}}(t,\mathbf{x})\,,$
(3)

Then, let us focus on eq. (1) in the sector in which heavy quarks with
momentum of order $\sqrt{MT}$ interact with gluons with energy of order $T$.
This would be equivalent to performing the matching between NRQCD and
$\textrm{NRQCD}_{sh}$ at tree level

$\mathcal{L}_{\xi}=\sum_{\mathbf{p}\neq
0}\xi^{\dagger}_{\mathbf{p}}\left\\{iD_{0}-\frac{(\mathbf{p}-i\boldsymbol{\nabla}+g\mathbf{A})^{2}}{2M}+c_{4}\frac{p^{4}}{8M^{3}}+c_{F}g\frac{\boldsymbol{\sigma}\cdot\mathbf{B}}{2M}\right\\}\xi_{\mathbf{p}}+\cdots$
(4)

It is more convenient to rearrange the terms in the following way

$\mathcal{L}_{\xi}=\sum_{\mathbf{p}\neq
0}\xi^{\dagger}_{\mathbf{p}}\left\\{iD_{0}-\frac{p^{2}}{2M}+i\frac{\mathbf{p}\cdot(\boldsymbol{\nabla}+ig\mathbf{A})}{M}+\frac{(\boldsymbol{\nabla}+ig\mathbf{A})^{2}}{2M}+c_{4}\frac{p^{4}}{8M^{3}}+c_{F}g\frac{\boldsymbol{\sigma}\cdot\mathbf{B}}{2M}\right\\}\xi_{\mathbf{p}}+\cdots$
(5)

The advantage of this equation compared to the previous one is that now each
term has a well-defined power counting. Since the only scale that has not been
integrated out is $T$, it follows that $\xi_{\mathbf{p}}$ is of size
$T^{3/2}$. Therefore, the first two terms in eq. (5) are of order $1$. The
third term is of order $\sqrt{\frac{T}{M}}$ and the rest of terms are of order
$\frac{T}{M}$. We have obtained eq. (5) by tree-level manipulations of the
NRQCD Lagrangian. However, it is easy to convince our selves that it
corresponds to the heavy quark sector of $\textrm{NRQCD}_{sh}$. The only
differences that a proper matching would bring up in this case are possible
sub-leading corrections to $M$, and the Wilson coefficients $c_{4}$ and
$c_{F}$. Note that Galilean symmetry and reparametrization invariance [36]
constraint the form of eq. (5).

Let us now mention some features of $\textrm{NRQCD}_{sh}$ we believe are worth
emphasizing. The first one is that, up to a trivial shift in the energy, the
field $\xi_{\mathbf{p}}$ behaves at leading-order as a static quark. This
implies that we can use $\textrm{NRQCD}_{sh}$ to compute properties of a heavy
quark with a momentum of order $\sqrt{MT}$ by performing perturbations around
the static case. The second remarkable feature that we want to comment is that
the study of the Wigner distribution is very much simplified in
$\textrm{NRQCD}_{sh}$. The Wigner distribution of a heavy quark in NRQCD takes
the following form

$f_{W}(\mathbf{p},\mathbf{R})=\int\,d^{3}r\mathrm{e}^{-i\mathbf{p}\mathbf{r}}\mathit{Tr}\left(\psi^{\dagger}\left(t,\mathbf{R}-\frac{\mathbf{r}}{2}\right)\phi\left(t:\mathbf{R}-\frac{\mathbf{r}}{2},\mathbf{R}+\frac{\mathbf{r}}{2}\right)\psi\left(t,\mathbf{R}+\frac{\mathbf{r}}{2}\right)\rho\right)\,,$
(6)

where $\phi$ is a Wilson line introduced in the definition such that the
Wigner distribution is gauge invariant [37]. By virtue of this Wilson line,
the $\mathbf{p}$ momentum in the Wigner distribution corresponds to the
kinetic momentum and not to the canonical momentum conjugate [37]. We are
interested in the case in which $p\sim\sqrt{MT}$ and the dependence of $f_{W}$
with $R$ is only sizeable over distances much larger than $1/T$. In this case,
$f_{W}$ is matched in $\textrm{NRQCD}_{sh}$ into

$\begin{split}f_{W}(\mathbf{p},\mathbf{R})=Z_{0}(\mathbf{r})\mathit{Tr}\left(\xi^{\dagger}_{\mathbf{p}}(t,\mathbf{R})\xi_{\mathbf{p}}(t,\mathbf{R})\rho\right)\\\
+iZ_{1}(\mathbf{r})\mathit{Tr}\left(\mathbf{r}\cdot\mathbf{D}\xi^{\dagger}_{\mathbf{p}}(t,\mathbf{R})\xi_{\mathbf{p}}(t,\mathbf{R})-\mathbf{r}\cdot\xi^{\dagger}_{\mathbf{p}}(t,\mathbf{R})\mathbf{D}\xi_{\mathbf{p}}(t,\mathbf{R})\rho\right)\\\
+Z_{2}(\mathbf{r})r\mathit{Tr}\left(\xi^{\dagger}_{\mathbf{p}}(t,\mathbf{R})\mathbf{r}\cdot\mathbf{E}(t,\mathbf{R})\xi_{\mathbf{p}}(t,\mathbf{R})\rho\right)+\cdots\end{split}$
(7)

This equation is valid up to terms smaller than $\frac{T}{M}$. Note that,
since both the lhs and the rhs can be understood as a pseudo-probability
distribution normalized to $1$, $Z_{0}(\mathbf{r})=1+\mathcal{O}(r^{2}T^{2})$.
In conclusion, up to sub-leading corrections, we can identify
$\mathit{Tr}\left(\xi^{\dagger}_{\mathbf{p}}(t,\mathbf{R})\xi_{\mathbf{p}}(t,\mathbf{R})\rho\right)$
with the Wigner distribution. The use of $\textrm{NRQCD}_{sh}$ allows seeing
the Wigner distribution as a probability distribution (at least at leading
order) encoded in a local operator.

Let us now discuss some properties of the propagators in $\textrm{NRQCD}_{sh}$
in thermal field theory that will be useful in the following. Now on, and in
order to simplify the notation, we will drop the sub-index $\mathbf{p}$ from
the field $\xi$. First, let us discuss the dilute limit, which we define as
the limit in which

$-\mathit{Tr}(\xi(t,\mathbf{r}_{1})\xi^{\dagger}(t,\mathbf{r}_{2})\rho)\gg\mathit{Tr}(\xi^{\dagger}(t,\mathbf{r}_{2})\xi(t,\mathbf{r}_{1})\rho)\,.$
(8)

In this limit,

$S_{11}(t,\mathbf{r})=\mathit{Tr}(\mathcal{P}\xi_{1}(t,\mathbf{r})\xi_{1}^{\dagger}(0,\mathbf{0})\rho)\sim\theta(t)\mathit{Tr}(\xi(t,\mathbf{r})\xi^{\dagger}(0,\mathbf{0})\rho)\,,$
(9)
$S_{22}(t,\mathbf{r})=\mathit{Tr}(\mathcal{P}\xi_{2}(t,\mathbf{r})\xi_{2}^{\dagger}(0,\mathbf{0})\rho)\sim\theta(-t)\mathit{Tr}(\xi(t,\mathbf{r})\xi^{\dagger}(0,\mathbf{0})\rho)\,,$
(10)
$S_{12}(t,\mathbf{r})=S^{<}(t,\mathbf{r})=\mathit{Tr}(\mathcal{P}\xi_{1}(t,\mathbf{r})\xi_{2}^{\dagger}(0,\mathbf{0})\rho)=-\mathit{Tr}(\xi^{\dagger}(0,\mathbf{0})\xi(t,\mathbf{r}))\rho)\sim
0\,,$ (11)

and

$S_{21}(t,\mathbf{r})=S^{>}(t,\mathbf{r})=\mathit{Tr}(\mathcal{P}\xi_{2}(t,\mathbf{r})\xi_{1}^{\dagger}(0,\mathbf{0})\rho)\sim\mathit{Tr}(\xi(t,\mathbf{r})\xi^{\dagger}(0,\mathbf{0})\rho)\,.$
(12)

We note that these results are true as long as we are exactly in the dilute
limit. It is also useful to write the expression in the Keldysh representation
[38].

$S_{R}(t,\mathbf{r})=\theta(t)\mathit{Tr}(\\{\xi(t,\mathbf{r})\xi^{\dagger}(0,\mathbf{0})\\}\rho)\sim
S_{11}(t,\mathbf{r})\,,$ (13)
$S_{A}(t,\mathbf{r})=-\theta(-t)\mathit{Tr}(\\{\xi(t,\mathbf{r})\xi^{\dagger}(0,\mathbf{0})\\}\rho)\sim-
S_{22}(t,\mathbf{r})\,,$ (14)

and

$S_{S}(t,\mathbf{r})=\mathit{Tr}([\xi(t,\mathbf{r})\xi^{\dagger}(0,\mathbf{0})]\rho)\sim
S_{21}(t,\mathbf{r})\,.$ (15)

The following useful relation is exactly fulfilled in the dilute limit

$S_{S}(t,\mathbf{r})\sim S_{R}(t,\mathbf{r})-S_{A}(t,\mathbf{r})\,.$ (16)

This implies that in the dilute limit the retarded propagator contains all the
relevant information, since the advanced propagator is the complex conjugate
of the retarded.

Let us now consider that the density of heavy quarks is small but non-zero. In
this case, $S^{<}$ is small but not zero. Since this propagator is directly
related with the distribution of heavy quarks, finite density corrections in
other propagators have a sub-leading impact on the mentioned distribution.

Finally, let us write the heavy quark propagator in $\textrm{NRQCD}_{sh}$ at
tree level. The tree level propagators are more compactly written in the
Keldysh representation

$S^{0}_{R}(t,\mathbf{r})=\int\frac{\,d^{4}k}{(2\pi)^{4}}\mathrm{e}^{-ik_{0}t+i\mathbf{k}\mathbf{r}}\frac{i}{k_{0}-\frac{p^{2}}{2M}+i\epsilon}=\theta(t)\mathrm{e}^{-\frac{ip^{2}t}{2M}}\delta^{(3)}(\mathbf{r})\,,$
(17)
$S^{0}_{A}(t,\mathbf{r})=\int\frac{\,d^{4}k}{(2\pi)^{4}}\mathrm{e}^{-ik_{0}t+i\mathbf{k}\mathbf{r}}\frac{i}{k_{0}-\frac{p^{2}}{2M}-i\epsilon}=-\theta(-t)\mathrm{e}^{-\frac{ip^{2}t}{2M}}\delta^{(3)}(\mathbf{r})\,,$
(18)

and

$\begin{split}S^{0}_{S}(t,\mathbf{r},\mathbf{R})=\mathit{Tr}([\xi^{0}\left(t,\mathbf{R}+\frac{\mathbf{r}}{2}\right){\xi^{0}}^{\dagger}\left(0,\mathbf{R}-\frac{\mathbf{r}}{2}\right)]\rho)\\\
=\int\frac{\,d^{4}k}{(2\pi)^{4}}\mathrm{e}^{-ik_{0}t+i\mathbf{k}\mathbf{r}}2\pi\delta\left(k_{0}-\frac{p^{2}}{2M}\right)(1-2f^{0}(\mathbf{p}+\mathbf{k},\mathbf{R}))\,,\end{split}$
(19)

where the $0$ super-index denotes tree level quantities and we have assumed
that the tree level distribution function of heavy quarks $f^{0}$ is a very
smooth function over distances of size $\mathbf{r}$. Note also that to be
consistent with the $\textrm{NRQCD}_{sh}$ power counting, the function $f^{0}$
(or the resummed equivalent that we will introduce later) must be expanded

$f^{0}(\mathbf{p}+\mathbf{k},\mathbf{R})\sim
f^{0}(\mathbf{p},\mathbf{R})+\mathbf{k}\boldsymbol{\nabla}_{\mathbf{p}}f^{0}(\mathbf{p},\mathbf{R})+\frac{k^{i}k^{j}}{2}\Delta^{ij}_{\mathbf{p}}f^{0}(\mathbf{p},\mathbf{R})+\cdots$
(20)

This implies that

$\begin{split}S^{0}_{S}(t,\mathbf{r},\mathbf{R})\sim\mathrm{e}^{-\frac{ip^{2}t}{2M}}\left[(1-2f^{0}(\mathbf{p},\mathbf{R}))\delta^{(3)}(\mathbf{r})+2\boldsymbol{\nabla}\delta^{(3)}(\mathbf{r})\boldsymbol{\nabla}_{\mathbf{p}}f^{0}(\mathbf{p},\mathbf{R})\right.\\\
\left.-\Delta^{ij}\delta^{(3)}(\mathbf{r})\Delta^{ij}_{\mathbf{p}}f^{0}(\mathbf{p},\mathbf{R})+\cdots\right]\,.\end{split}$
(21)

Finally, let us mention that the Feynmann rules for $\textrm{NRQCD}_{sh}$ can
be found in Appendix A.

##  III Langevin Effective Theory

In this section we introduce Langevin Effective Theory (LET). It is obtained
from $\textrm{NRQCD}_{sh}$ after integrating out the scale $T$. In this case
we are dealing with heavy quarks with momentum $\mathbf{p}+\mathbf{k}$ where
now the residual momentum $k$ is much smaller than the temperature. Note that
we can always redefine $p$ such that this relation is fulfilled. In order to
construct LET, we follow the observations of [18]. The influence functional of
LET can be written as,

$S_{LET}=S_{1}-S_{2}+\mathcal{I}+S_{HTL}\,,$ (22)

where $S_{1}$ ($S_{2}$) is the piece of the action that involves only heavy
quark fields of type $1$ ($2$). $\mathcal{I}$ is a new type of contribution in
which heavy quark fields of both type $1$ and $2$ appear. Finally, $S_{HTL}$
is the well-known Hard Thermal Loop action [39]. If the scale $T$ did not
induce any dissipative effects, then $\mathcal{I}$ would be zero and $S_{2}$
would be equal to $S_{1}$ just changing the fields of type $2$ by fields of
type $1$.

The construction of LET is substantially simplified by applying the dilute
limit. We know that, in this limit, we can ignore the doubling of degrees of
freedom when computing Green functions in which only heavy fields of type $1$
(or $2$) appear [7, 8]. Moreover, in the dilute limit and as long as the heavy
particle is non-relativistic in the frame in which the medium is at rest, the
symmetries of $\textrm{NRQCD}_{sh}$ are not broken by the presence of the
medium. Therefore, the form of $S_{1}$ is equal to the action of
$\textrm{NRQCD}_{sh}$. However, the Wilson coefficients are different. In
fact, some of these Wilson coefficients can be complex. Then,

$\begin{split}S_{1}=\int\,d^{4}x\xi^{\dagger}_{1}\left[(1-\delta
Z)\left(iD_{0}-\frac{p^{2}}{2M}\right)-\delta
E+i\frac{\Gamma}{2}+i\frac{\mathbf{p}\cdot(\boldsymbol{\nabla}+ig\mathbf{A})}{M}+\frac{(\boldsymbol{\nabla}+ig\mathbf{A})^{2}}{2M}\right.\\\
\left.+\tilde{c}_{4}\frac{p^{4}}{8M^{3}}+\tilde{c}_{F}g\frac{\boldsymbol{\sigma}\cdot\mathbf{B}}{2M}\right]\xi_{1}+\cdots\end{split}$
(23)

where we allow both $\delta Z$, $\delta E$ and $\Gamma$ to be polynomials
involving $M$, $T$ and $p^{2}$. Note that $\delta Z$ can have both a real and
an imaginary part. The real part of $\delta Z$ can be reabsorbed by a unitary
transformation. Whether the imaginary part of $\delta Z$ can be reabsorbed by
a transformation of the fields is beyond the scope of this work. We note that,
since our main focus is the study of $\frac{df}{dt}$, we can ignore a non-zero
value of $\delta Z$. The reason is that the wave-function renormalization is
not a secular effect, meaning that its effects on the evolution on the
distribution of heavy quarks does not become larger as we study longer times.
This is in contrast to what happens to corrections to the mass and the decay
width. For example, even if the decay width is small, it becomes a leading
order effect at large enough times.

$S_{2}$ can be obtained from $S_{1}$ by changing the fields of type $1$ to
fields of type $2$ and by making the complex conjugate of the Wilson
coefficients. In our case, this means making the changes $\delta Z\to\delta
Z^{*}$ and $i\frac{\Gamma}{2}\to-i\frac{\Gamma}{2}$. We note also, that since
$\delta E$ and $\Gamma$ are obtained by performing a matching computation to
$\textrm{NRQCD}_{sh}$, they are polynomials with the following structure

$\delta
E=\alpha^{0}T+\alpha^{1}\frac{T^{2}}{M}+\alpha^{2}\frac{p^{2}T}{M^{2}}+\mathcal{O}\left(\frac{T^{3}}{M^{2}}\right)\,,$
(24)
$\Gamma=\beta^{0}T+\beta^{1}\frac{T^{2}}{M}+\beta^{2}\frac{p^{2}T}{M^{2}}+\mathcal{O}\left(\frac{T^{3}}{M^{2}}\right)\,.$
(25)

Regarding $\mathcal{I}$, it can be fixed by imposing the following conditions:

  * •

$S_{LET}$ is equal to zero if fields of type $1$ are equal to fields of type
$2$ [18].

  * •

In the dilute limit, the propagator $S_{12}$ is zero.

Using this, we get

$\mathcal{I}=\int\,d^{4}x\xi^{\dagger}_{2}\left[2i\mathrm{Im}Z\left(iD_{0}-\frac{p^{2}}{2M}\right)-i\Gamma-i\mathrm{Im}\tilde{c}_{4}\frac{p^{4}}{4M^{3}}-i\mathrm{Im}\tilde{c}_{F}\frac{\boldsymbol{\sigma}\cdot\mathbf{B}}{M}\right]\xi_{1}+\cdots$
(26)

Let us now discuss the issue of gauge invariance. The equations we wrote are
only invariant regarding transformations in which the fields are modified in
the same in both branches of the Schwinger-Keldysh contour. More specifically,
there is an explicit invariance under the following type of infinitesimal
gauge transformations:

$\begin{split}\xi_{1}\to\xi_{1}+ig\Lambda\xi_{1}\,,\\\
\xi_{2}\to\xi_{2}+ig\Lambda\xi_{2}\,,\\\ A^{\mu}_{1}\to
A^{\mu}_{1}+\partial^{\mu}\Lambda+ig[\Lambda,A^{\mu}_{1}]\,,\\\ A^{\mu}_{2}\to
A^{\mu}_{2}+\partial^{\mu}\Lambda+ig[\Lambda,A^{\mu}_{2}]\,.\end{split}$ (27)

Apparently, something has been missed in going from $\textrm{NRQCD}_{sh}$ to
LET. $\textrm{NRQCD}_{sh}$ is invariant under transformations in which each
branch of the Schwinger-Keldysh contour is independently transformed with a
different $\Lambda_{i}$. How to recover this more general invariance in EFTs
where terms mixing the two branches appear was discussed in [19, 20, 21],
however the solution the found is not suitable for our case. We have proposed
an alternative solution in appendix B, where we discuss this issue and its
solution in more detail. However, at the end of the day, this issue has little
practical importance for the computation at hand and will only complicate the
notation. Careful readers might have notived that we also did not mention
whether the gauge fields entering $D_{0}$ and $\mathbf{B}$ in eq. (26) are of
type $1$ or $2$. More details about this are also given in appendix B.

The previous way of presenting the action of $S_{LET}$ is useful to perform
the matching to the full theory. However, we might rearrange the contributions
in a more physically meaningful way.

$S_{LET}=\tilde{S}_{1}-\tilde{S}_{2}+\mathcal{D}\,,$ (28)

where

$\begin{split}\tilde{S}_{i}=\int\,d^{4}x\xi^{\dagger}_{i}\left[(1-\mathrm{Re}\delta
Z)\left(iD_{0}-\frac{p^{2}}{2M}\right)-\delta
E+i\frac{\mathbf{p}\cdot(\boldsymbol{\nabla}+ig\mathbf{A})}{M}+\frac{(\boldsymbol{\nabla}+ig\mathbf{A})^{2}}{2M}\right.\\\
\left.+\mathrm{Re}\tilde{c}_{4}\frac{p^{4}}{8M^{3}}+\mathrm{Re}\tilde{c}_{F}g\frac{\boldsymbol{\sigma}\cdot\mathbf{B}}{2M}\right]\xi_{i}+\cdots\end{split}$
(29)

is the unitary part of the evolution and $\mathcal{D}$ is the dissipative part

$\begin{split}\mathcal{D}=i\int\,d^{4}x\left(\begin{array}[]{cc}\xi^{\dagger}_{1}&\xi^{\dagger}_{2}\end{array}\right)\left[-\mathrm{Im}\delta
Z\left(iD_{0}-\frac{p^{2}}{2M}\right)+\frac{\Gamma}{2}+\mathrm{Im}\tilde{c}_{4}\frac{p^{4}}{8M^{3}}+\mathrm{Im}\tilde{c}_{F}\frac{\boldsymbol{\sigma}\cdot\mathbf{B}}{2M}\right]\\\
\otimes\left(\begin{array}[]{cc}1&0\\\
-2&1\end{array}\right)\left(\begin{array}[]{c}\xi_{1}\\\
\xi_{2}\end{array}\right)\,.\end{split}$ (30)

It might be illustrative to write $S_{LET}$ in the Keldysh basis [38, 40]. Let
us introduce

$\xi_{Cl}=\frac{\xi_{1}+\xi_{2}}{2}\,,$ (31)

which we call the classical field and

$\xi_{Q}=\xi_{1}-\xi_{2}\,,$ (32)

the quantum field. There are two remarkable properties that can be seen just
introducing this basis. First, the condition that $S_{LET}$ is zero when
fields of type $1$ are equal of type $2$ can be rephrased as imposing that
$S_{LET}=0$ when $\xi_{Q}=0$. It also follows that a QFT describing a closed
system (without dissipation) only has terms containing an odd number of
quantum fields. However, in an EFT obtained after integrating out medium
degrees of freedom (such as the one we are studying in this paper), we might
have terms containing an even number of quantum fields. In summary, terms with
an even number of quantum fields are forbidden in $\tilde{S}_{i}$ while they
are allowed in $\mathcal{D}$. To see this more explicitly, let us introduce
the following operators:

$\begin{split}\Delta_{S}=(1-\mathrm{Re}\delta
Z)\left(iD_{0}-\frac{p^{2}}{2M}\right)-\delta
E+i\frac{\mathbf{p}\cdot(\boldsymbol{\nabla}+ig\mathbf{A})}{M}+\frac{(\boldsymbol{\nabla}+ig\mathbf{A})^{2}}{2M}\\\
+\mathrm{Re}\tilde{c}_{4}\frac{p^{4}}{8M^{3}}+\mathrm{Re}\tilde{c}_{F}g\frac{\boldsymbol{\sigma}\cdot\mathbf{B}}{2M}\,,\end{split}$
(33)

and

$\Delta_{D}=-\mathrm{Im}\delta
Z\left(iD_{0}-\frac{p^{2}}{2M}\right)+\frac{\Gamma}{2}+\mathrm{Im}\tilde{c}_{4}\frac{p^{4}}{8M^{3}}+\mathrm{Im}\tilde{c}_{F}\frac{\boldsymbol{\sigma}\cdot\mathbf{B}}{2M}\,.$
(34)

Then, we can write $S_{LET}$ in the following way

$S_{LET}=\int\,d^{4}x\left(\xi^{\dagger}_{Cl}\Delta_{S}\xi_{Q}+\xi^{\dagger}_{Q}\Delta_{S}\xi_{Cl}+\xi^{\dagger}_{Q}\Delta_{D}\xi_{Cl}-\xi^{\dagger}_{Cl}\Delta_{D}\xi_{Q}+\xi^{\dagger}_{Q}\Delta_{D}\xi_{Q}\right)\,.$
(35)

Writing $S_{LET}$ in this way provides some extra insight. In a situation in
which $\xi_{Q}$ is suppressed, the leading contribution comes from terms
linear in $\xi_{Q}$. Integrating over $\xi_{Q}$ considering only these linear
terms gives a Dirac delta that forces $\xi_{Cl}$ to follow the classical
equations of motion. Terms quadratic in $\xi_{Q}$ can be seen as originating
from a classical random source, therefore we can understand them as
fluctuations [41].

Now, let us discuss the case in which we still consider that heavy quarks are
dilute but we take into account the first corrections proportional to their
density. We call this NLO dilute corrections. Regarding the symmetries of the
EFT, we consider that $f(\mathbf{p})$ does not have any preferred direction
other than $\mathbf{p}$ itself. Taking this into account, $\mathcal{D}$ has to
be modified in order to include an extra term

$\begin{split}\mathcal{D}=i\int\,d^{4}x\left(\begin{array}[]{cc}\xi^{\dagger}_{1}&\xi^{\dagger}_{2}\end{array}\right)\left[\left(-\mathrm{Im}\delta
Z\left(iD_{0}-\frac{p^{2}}{2M}\right)+\frac{\Gamma}{2}+\mathrm{Im}\tilde{c}_{4}\frac{p^{4}}{8M^{3}}+\mathrm{Im}\tilde{c}_{F}\frac{\boldsymbol{\sigma}\cdot\mathbf{B}}{2M}\right)\right.\\\
\left.\otimes\left(\begin{array}[]{cc}1&0\\\
-2&1\end{array}\right)+\Delta\Gamma\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right)\right]\left(\begin{array}[]{c}\xi_{1}\\\
\xi_{2}\end{array}\right)\,.\end{split}$ (36)

We note that the term that we have added is the leading one that we can add
that does not fulfill the condition that the $S_{12}$ propagator has to be
zero (dilute limit) but that fulfills the rest of conditions that we have
discussed in this section. Again, we can rewrite the action in the classical-
quantum basis,

$\begin{split}S_{LET}=\int\,d^{4}x\left(\xi^{\dagger}_{Cl}\Delta_{S}\xi_{Q}+\xi^{\dagger}_{Q}\Delta_{S}\xi_{Cl}+\xi^{\dagger}_{Q}\Delta_{D}\xi_{Cl}-\xi^{\dagger}_{Cl}\Delta_{D}\xi_{Q}+\xi^{\dagger}_{Q}\Delta_{D}\xi_{Q}\right.\\\
\left.+\xi^{\dagger}_{Cl}\Delta\Gamma\xi_{Q}-\xi^{\dagger}_{Q}\Delta\Gamma\xi_{Cl}\right)\,.\end{split}$
(37)

We note that all Wilson coefficients can be affected by the NLO dilute
corrections in a sub-leading way. The specific property of $\Delta\Gamma$ is
that it vanishes in the exact dilute limit.

There are additional symmetries in the EFT that impose relations between the
different terms in $\Gamma$ and $\Delta\Gamma$. The origin of these relations
is the Schwinger-Keldysh symmetry [19, 20, 21], also known as the fluctuation-
dissipation theorem. However, let us postpone this discussion until next
section, since it is very much related with the evolution of $f(\mathbf{p})$.

##  IV Evolution of $f(\mathbf{p})$ in LET

In this section, we show how to compute the evolution of $f(\mathbf{p})$ in
the EFT we introduced before. At the same time, this will also allow us to
introduce further constraints on the Wilson coefficients by imposing that the
fluctuation-dissipation theorem is fulfilled. On more physical terms, this
means the following. We are studying the case of a dilute distribution of
heavy quarks evolving in a large bath in thermal equilibrium at a temperature
such that $T\ll M$. Therefore, at very large times $f(\mathbf{p})$ must be
equal to the thermal distribution, $\mathrm{e}^{-\frac{p^{2}}{2MT}}$.

The information on the distribution of heavy quarks can be found more directly
in the $S^{<}(p)$ propagator, which, as we have seen before, goes to zero in
the dilute limit. In order to be more precise, we define the distribution
function $f$ such that

$S^{<}(k)=f(\mathbf{p}+\mathbf{k})(S_{R}(k)-S_{A}(k))\,,$ (38)

where, as usual, we are refering to the propagators of the field $\xi_{p}$. To
study this propagator, it is convenient to use the Kadanoff-Baym equations
[42]. A recent application of these equations in the context of heavy-ion
collisions can be found in [43]. These equations are deduced by performing a
Dyson-Schwinger type of resummation of the self-energies. In our case, we will
just perform a resummation of the tree-level self-energies as obtained
directly from the LET action.

$\begin{split}\Pi_{R}=\delta E-i\frac{\Gamma}{2}\,,\\\ \Pi_{A}=\delta
E+i\frac{\Gamma}{2}\,,\\\ \Pi^{<}=-i\Delta\Gamma\,.\end{split}$ (39)

We note that, when studying a system evolving in a plasma, some kind of
resummation is always needed to deal with secular effects. In other words,
small perturbations that grow with time need to be resummed because they
become leading order effects at large enough times.

The propagator $S^{<}$ is a function of two times,

$S^{<}(t_{1},\mathbf{r}_{1};t_{2},\mathbf{r}_{2})=\textit{Tr}(\xi_{1}(t_{1},\mathbf{r}_{1})\xi^{\dagger}(t_{2},\mathbf{r}_{2})\rho)\,.$
(40)

In thermal equilibrium and due to translational invariance it is only a
function of $\tau=t_{1}-t_{2}$. More generally, it is also a function of
$t=\frac{t_{1}+t_{2}}{2}$. In order to study the evolution of $f(\mathbf{p})$,
the more direct way is to look at the evolution of $S^{<}$ as a function of
$t$ for $\tau=0$. The Kadanoff-Baym equations give us the evolution on each
time separately

$\begin{split}\partial_{t_{1}}S^{<}=-i\left(\frac{p^{2}}{2M}+\Pi_{R}\right)S^{<}+i\Pi^{<}S_{A}\,,\\\
\partial_{t_{2}}S^{<}=i\left(\frac{p^{2}}{2M}+\Pi_{A}\right)S^{<}-i\Pi^{<}S_{R}\,.\\\
\end{split}$ (41)

From this, it follows that the evolution with $T$ is given by

$\partial_{t}S^{<}=-i(\Pi_{R}-\Pi_{A})S^{<}-i\Pi^{<}(S_{R}-S_{A})\,.$ (42)

Note that the spectral function $\rho$ is given by $\rho=S_{R}-S_{A}$, which
at tree level is $\rho_{0}=2\pi\delta\left(k_{0}-\frac{p^{2}}{2M}\right)$. We
are studying the case of a heavy particle interacting with a medium in thermal
equilibrium. At very large times we should arrive to a steady state in which
the distribution of heavy particles is also in thermal equilibrium. This means
that at late times

$S^{<}=-\frac{\Pi^{<}(S_{R}-S_{A})}{\Pi_{R}-\Pi_{A}}\,.$ (43)

At the same time, at thermal equilibrium the fluctuation-dissipation theorem
must be fulfilled

$S^{<}=f_{eq}(\mathbf{p}+\mathbf{k})(S_{R}-S_{A})\,,$ (44)

where $f_{eq}(p)=N\mathrm{e}^{-\frac{p^{2}}{2M}}$ is the thermal equilibrium
distribution function and $N$ a normalization factor. We obtain the relation

$\Pi^{<}(\mathbf{p},\mathbf{k},f_{eq})=f_{eq}(\mathbf{p}+\mathbf{k})(\Pi_{R}(\mathbf{p},\mathbf{k})-\Pi_{A}(\mathbf{p},\mathbf{k}))\,,$
(45)

that must be fulfilled. Note that we have explicitly written the dependency
with the momentum and the distribution function. To simplify, let us focus on
the case $\mathbf{k}=\mathbf{0}$ and write the expression in terms of the
Wilson coefficients of LET

$\Delta\Gamma\left(\mathbf{p},\mathrm{e}^{-\frac{p^{2}}{2M}}\right)=\mathrm{e}^{-\frac{p^{2}}{2M}}\Gamma(\mathbf{p})\,.$
(46)

$\Delta\Gamma$ is proportional to $f$ or its derivatives

$\Delta\Gamma(\mathbf{p},f_{eq})=\left(\gamma^{0}+\frac{\gamma^{1}}{M}+\frac{\gamma^{2}p^{2}}{M^{2}}\right)f_{eq}(\mathbf{p})+\frac{\gamma^{3}}{M}\mathbf{p}\nabla_{\mathbf{p}}(f_{eq}(\mathbf{p}))+\gamma^{4}\Delta_{\mathbf{p}}f_{eq}(\mathbf{p})+\mathcal{O}\left(\frac{T}{M}\right)^{3/2}\,.$
(47)

At the same time, a similar expansion is valid for $\Gamma$

$\Gamma=\beta^{0}+\frac{\beta^{1}}{M}+\frac{\beta^{2}p^{2}}{M^{2}}\,.$ (48)

Then, this implies that the following relations must be fulfilled:

$\begin{split}\gamma^{0}=\beta^{0}\,,\\\
\gamma^{1}-\frac{3\gamma^{4}}{T}=\beta^{1}\,,\\\
\gamma^{2}-\frac{\gamma^{3}}{T}+\frac{\gamma^{4}}{T^{2}}=\beta^{2}\,.\end{split}$
(49)

In order to compute $\frac{\partial f(\mathbf{p})}{\partial t}$, we can use
eqs. (42), (44) and (49) in the case $\mathbf{k}=\mathbf{0}$

$\frac{\partial f(\mathbf{p})}{\partial t}=-\Gamma f(\mathbf{p})+\Delta\Gamma
f(\mathbf{p})=\left(\frac{3}{M}-\frac{p^{2}}{M^{2}T}\right)\left(\frac{\gamma^{4}}{T}-\gamma^{3}\right)f(\mathbf{p})+\frac{\gamma^{3}}{M}\mathbf{\nabla}(\mathbf{p}f(\mathbf{p}))+\gamma^{4}\Delta_{\mathbf{p}}f(\mathbf{p})\,.$
(50)

Another condition that must be fulfilled is unitarity. This implies that

$\frac{\partial}{\partial t}\int\,d^{3}pf(\mathbf{p})=0\,.$ (51)

This imposes an additional condition, this is $\gamma^{4}=T\gamma^{3}$. Note
that what we obtain is the evolution of $f(\mathbf{p})$ that corresponds to a
Fokker-Planck equation, in which the drag coefficient is given by
$\frac{\gamma^{3}}{M}$.

In summary, imposing all the constraints coming from unitarity and the
fluctuation-dissipation theorem, we get the following expression for
$\Delta\Gamma$

$\Delta\Gamma(\mathbf{p},f)=\left(\beta^{0}+\frac{\beta^{1}}{M}+\frac{\beta^{2}p^{2}}{M^{2}}\right)f(\mathbf{p})+\frac{\kappa}{2MT}\nabla_{\mathbf{p}}(\mathbf{p}f(\mathbf{p}))+\frac{\kappa}{2}\Delta_{\mathbf{p}}f(\mathbf{p})+\mathcal{O}\left(\frac{T}{M}\right)^{3/2}\,,$
(52)

where $\kappa$ is the heavy quark diffusion coefficient [44]. We see that the
matching between $\mathrm{NRQCD}_{sh}$ and $LET$ gets substantially
simplified. To determine $\Delta\Gamma$ at the order we are interested, we
only need to know $\Gamma$ in the dilute limit and the value of the heavy
quark diffusion coefficient. Moreover, if we are only interested in the
evolution of $f(\mathbf{p})$ we only need to know $\kappa$, as it was
expected, and we can see by writing the evolution of $f(\mathbf{p})$
considering all the constraints

$\frac{\partial f(\mathbf{p})}{\partial
t}=\frac{\kappa}{2MT}\mathbf{\nabla}(\mathbf{p}f(\mathbf{p}))+\frac{\kappa}{2}\Delta_{\mathbf{p}}f(\mathbf{p})\,.$
(53)

##  V Matching between $\mathrm{NRQCD}_{sh}$ and LET

In this section, we are going to perform the matching between
$\mathrm{NRQCD}_{sh}$ and LET in the one gluon exchange approximation. The
reader might wonder why we do not compute the matching at a given order in
perturbation theory instead. The reason is related to the special properties
of gauge theories at finite temperature. A one-loop matching will lead to
$\kappa=0$, meaning that $f(\mathbf{p})$ does not change. On the other hand, a
two-loop matching would involve a complex computation of $\Gamma$ that has
little phenomenological impact due to the symmetries that we discussed in the
previous section. The one gluon exchange approximation has the virtue of
giving a finite contribution to all Wilson coefficients already at leading
order, and we can regard it as an intermediate step between a one-loop and a
two-loop matching.

The strategy to perform the matching is going to be the following. First, we
are going to compute $\Pi_{R}$ in $\mathrm{NRQCD}_{sh}$, from this we can
obtain $\delta M$ and $\Gamma$. Second, we are going to compute $\Pi^{<}$, but
only the piece proportional to $\mathbf{\nabla}_{\mathbf{p}}f(\mathbf{p})$,
since we have seen in section IV that this is enough to fix all the relevant
Wilson coefficients. Regarding the computation of $\Pi_{R}$, we are going to
use the Keldysh basis following the graphical notation of [45]111Propagators
with a single arrow pointing to the right (left) are retarded (advanced).
Propagators with two outgoing arrows and a capacitor symbol are symmetric. The
rest of the symbols are defined in appendix A. For the computation of
$\Pi^{<}$, we find it more convenient to use the $1-2$ basis. We note that in
this section we are going to use the Feynmann rules discussed in Appendix A.

Figure 1: Diagrams contribution to $\Pi_{R}$ at order $T$ in
$\mathrm{NRQCD}_{sh}$. The computation is done in the Keldysh basis.

Let us begin by computing the leading contribution to $\Pi_{R}$ in
$\mathrm{NRQCD}_{sh}$, in other words, the terms that in the EFT power
counting are of order $T$. In fig. 1 we show the relevant diagrams. If we
write $\delta E$ as

$\delta E=\alpha^{0}+\frac{\alpha^{1}}{M}+\frac{\alpha^{2}p^{2}}{M^{2}}\,,$
(54)

and we use the notation of eq. (48), then the contribution from the diagrams
in fig. 1 is

$\begin{split}\alpha^{0}=-\frac{ig^{2}C_{F}}{4}\int\frac{\,d^{3}q}{(2\pi)^{3}}(\Delta_{00,R}+\Delta_{00,A})(0,\mathbf{q})\,,\\\
\beta^{0}=2Tg^{2}C_{F}\int\frac{\,d^{3}q}{(2\pi)^{3}}\rho_{00}^{\prime}(0,\mathbf{q})\,,\end{split}$
(55)

where $\Delta_{00}$ is the temporal gluon propagator and $\rho^{\prime}$ is
the first derivative over the temporal component of the spectral function of
the temporal gluon propagator [16]. Our computation is performed in the
Coulomb gauge.

Figure 2: Diagrams contribution to $\Pi_{R}$ at order $T\sqrt{\frac{T}{M}}$ in
$\mathrm{NRQCD}_{sh}$.

Next, let us discuss possible corrections to $\delta E$ and $\Gamma$ of order
$T\sqrt{\frac{T}{M}}$. By symmetry arguments, there should not be any.
However, let us check explicitly that it is the case. The relevant diagrams
are shown in fig. 2. Each circled correction to the propagator introduces a
factor $\frac{\mathbf{p}\mathbf{q}}{M}$, where $\mathbf{q}$ is the spatial
momentum of the temporal gluon. We can take $\mathbf{p}$ out of the integrand,
and since the medium does not have any preferred direction in space, the
result of the integration can only be zero.

Figure 3: Diagrams contribution to $\alpha^{1}$ and $\beta^{1}$ in
$\mathrm{NRQCD}_{sh}$.

Regarding the diagrams contributing at order $\frac{T^{2}}{M}$, we can divide
them in two classes. Those that contribute to $\alpha^{1}$ and $\beta^{1}$,
and those contributing to $\alpha^{2}$ and $\beta^{2}$. The first class is
shown in fig. 3. They give the following contribution:

$\begin{split}\alpha^{1}=\frac{g^{2}C_{F}}{4}\int\frac{\,d^{4}q}{(2\pi)^{4}}\Delta_{ii,S}(q)+\frac{g^{2}C_{F}}{8}\int\frac{\,d^{4}q}{(2\pi)^{4}}q^{2}\left(\frac{1}{(q_{0}-i\epsilon)^{2}}+\frac{1}{(q_{0}+i\epsilon)^{2}}\right)\Delta_{00,S}(q)\\\
-\frac{ig^{2}C_{F}}{8}\int\frac{\,d^{3}q}{(2\pi)^{3}}q^{2}\frac{d(\Delta^{R}_{00}+\Delta^{A}_{00})}{dq_{0}}(0,\mathbf{q})\,,\\\
\beta^{1}=-\frac{g^{2}C_{f}}{4}\int\frac{\,d^{3}q}{(2\pi)^{3}}q^{2}\rho^{\prime}_{00}(0,\mathbf{q})\,.\end{split}$
(56)

Figure 4: Diagrams contribution to $\alpha^{2}$ and $\beta^{2}$ in
$\mathrm{NRQCD}_{sh}$.

Diagrams contributing to $\alpha^{2}$ and $\beta^{2}$ are shown in fig. 4.
They result in the following contribution:

$\begin{split}\alpha^{2}=-i\frac{g^{2}C_{F}}{12}\int\frac{\,d^{3}q}{(2\pi)^{3}}(\Delta_{ii,R}+\Delta_{ii,A})\,,\\\
\beta^{2}=\frac{g^{2}C_{F}T}{6}\int\frac{\,d^{3}q}{(2\pi)^{3}}\rho_{ii}^{\prime}(0,\mathbf{q})-\frac{g^{2}}{6}\int\frac{\,d^{4}q}{(2\pi)^{4}}\left(\frac{i}{q_{0}+i\epsilon}\right)^{3}\Delta_{00,S}(q)\mathbf{q}^{2}\,.\end{split}$
(57)

211

212

Figure 5: Diagrams involved in the matching of $\kappa$.

Let us now compute $\Pi^{<}$ in $\mathrm{NRQCD}_{sh}$ in order to perform the
matching. We are going to focus on the piece proportional to
$\mathbf{p}\mathbf{\nabla}_{\mathbf{p}}f(\mathbf{p})$ since, as we discussed
previously, it is the only extra term needed to perform the matching. To
obtain this, we need to go to second order in the expansion shown in eq. (21).
Moreover, we have to take from the $\mathrm{NRQCD}_{sh}$ Lagrangian the
interaction that goes like
$\frac{\xi^{\dagger}\mathbf{p}\mathbf{\nabla}\xi}{M}$. Therefore, we need to
focus on the diagram shown in fig. 5. Note that for the computation of
$\Pi^{<}$ we are using the $1-2$ basis instead of the Schwinger-Keldysh one.
From this matching, we obtain

$\kappa=\frac{g^{2}C_{F}T}{3}\int\frac{\,d^{3}q}{(2\pi)^{3}}q^{2}\rho^{\prime}_{00}(0,\mathbf{q})\,.$
(58)

We can check that this result is compatible with the perturbative computation
of $\kappa$ in the Coulomb gauge [44, 47, 24]. Note that in the $q_{0}\to 0$
limit we can approximate $\mathbf{E}$ by $-\mathbf{\nabla}A_{0}$, as long as
we are not using the unitary gauge. It is also important to take into account
that we are making a small abuse of language by identifying the transport
coefficient $\kappa$ with the Wilson coefficient of LET. Strictly speaking,
the Wilson coefficient only includes the contribution to $\kappa$ from the
scale we are integrating out, in this case $T$.

21 Figure 6: An alternative diagram to consider in order to match $\kappa$.

With this we have finished the matching to the order of interest. However, for
the sake of cross-checking some of the relations we discussed in the previous
section, we are going to discuss an alternative way to match the value of
$\kappa$. This can be done looking at the diagram in fig. 6. However, now we
have to take the third term in eq. (21) to get the contribution proportional
to $\Delta f(\mathbf{p})$. Doing this, we can check that we get exactly the
same value of $\kappa$. There is even a third way of computing $\kappa$ which
is to look at the terms in $\Delta\Gamma$ that go like
$\frac{f(\mathbf{p})}{M}$ and subtract $\beta^{1}$. However, we are not going
to discuss further this way of doing the matching of $\kappa$.

Now we are ready to discuss how to derive a Fokker-Planck equation for heavy
quarks in QCD including the effects of the scales $T$ and $gT$. The first step
has already been discussed. We match $\mathrm{NRQCD}_{sh}$ to LET to include
the effect of the scale $T$ in the Wilson coefficients. The second step is to
compute $\Pi_{R}$ and $\Pi^{<}$ in LET including the effects of the scale
$gT$. Note however that the same symmetries and cancellations apply now, and
therefore, we only need to compute the contribution of the scale $gT$ to
$\kappa$. This can be obtained by computing the diagram in fig. 5, but now the
loop has to be computed using the HTL gluon propagator. Proceeding in this
way, we obtain the perturbative estimate of $\kappa$. It corresponds to eq.
(58) but now using the two loop perturbative result for $q\sim T$ and the HTL
approximation for $q\sim m_{D}$ [47, 24].

##  VI Heavy quarks with momentum $p\sim T$

In this section, we discuss the case of a heavy quark with a tri-momentum of
order $T$. In this case, we will proceed as follows:

  * •

As starting point, we will use NRQCD Lagrangian.

  * •

We will integrate out the scale $T$. After this we will arrive again to LET.
However, now the Wilson coefficients and the power counting will be slightly
different, since now $p$ is of order $T$ instead of $\sqrt{MT}$ as before.

  * •

In this case, $\Delta\Gamma$ will have a more involved dependency with $p$. As
a result, instead of a Fokker-Planck equation, we get an evolution similar to
a Boltzmann equation.

Let us start looking at LET. The Lagrangian is again formed by combining eq.
(23) and eq. (26) in the same way as before. The main difference is the value
of the Wilson coefficients.

Regarding $\delta E$ and $\Gamma$, it is easy to check that the results are
exactly the same as in the case $p\sim\sqrt{MT}$. The reason is that the
diagrams that appear doing the respective matchings in $\mathrm{NRQCD}_{sh}$
(for $p\sim\sqrt{MT}$) and in NRQCD (for $p\sim T$) are the same at any order
in the coupling constant, the only difference being the relative size of each
term. However, the situation is different for $\Delta\Gamma$, the reason is
that now this Wilson coefficient is, in general, a non-trivial function of $f$
and $p$ and we can not assume a polynomial form like in eq. (52). However, we
can assume that it is a polynomial in inverse powers of $1/M$ from the
structure of NRQCD. Then, we can write

$\Delta\Gamma(\mathbf{p},f)=\int\frac{\,d^{3}q}{(2\pi)^{3}}\left(C_{1}(q)+\frac{C_{2}(q)q^{2}}{M}+\frac{C_{3}(q)(2\mathbf{q}\mathbf{p}-q^{2})}{M}\right)f(\mathbf{p}-\mathbf{q})\,.$
(59)

There are several constraints that can be imposed to $C_{i}$. Eq. (46) must be
fulfilled, but now we need to take into account that we need to expand
$\mathrm{e}^{-\frac{p^{2}}{2MT}}$ to the order desired since
$\frac{p^{2}}{2MT}\ll 1$. This implies that

$\begin{split}\int\frac{\,d^{3}q}{(2\pi)^{3}}C_{1}(q)=\beta_{0}\,,\\\
\int\frac{\,d^{3}q}{(2\pi)^{3}}q^{2}\left(C_{2}(q)-C_{3}(q)-\frac{C_{1}(q)}{2T}\right)=\beta_{1}\,,\end{split}$
(60)

Additional constraints come from unitarity. We can repeat the arguments of
section IV to arrive to

$\partial_{t}f(\mathbf{p})=\Delta\Gamma(\mathbf{p},f)-\Gamma f(\mathbf{p})\,.$
(61)

Imposing that $\partial_{t}\int\frac{\,d^{3}p}{(2\pi)^{3}}f(\mathbf{p}=0)$ we
deduce that

$\int\frac{\,d^{3}p}{(2\pi)^{3}}\Delta\Gamma(\mathbf{p},f)=\Gamma\,.$ (62)

Starting from the previous equation and using the change of variables
$\mathbf{p}\to\mathbf{p}+\mathbf{q}$ we obtain the additional constraint

$\int\frac{\,d^{3}q}{(2\pi)^{3}}q^{2}\left(C_{2}(q)+C_{3}(q)\right)=\beta_{1}\,.$
(63)

Using the previous equation and some trivial manipulations, we arrive at our
final form of the evolution equation

$\begin{split}\partial_{t}f(\mathbf{p})=\int\frac{\,d^{3}q}{(2\pi)^{3}}\left(C_{1}(q)+\frac{C_{2}(q)q^{2}}{M}\right)(f(\mathbf{p}-\mathbf{q})-f(\mathbf{p}))\\\
+\frac{1}{M}\int\frac{\,d^{3}q}{(2\pi)^{3}}C_{3}(q)(2\mathbf{q}\mathbf{p}-q^{2})(f(\mathbf{p}-\mathbf{q})+f(\mathbf{p}))\,.\end{split}$
(64)

As an illustration, let us discuss the perturbative matching. This can be done
by looking at the diagrams in figs. 6 and 5, but now taking into account that
the expansion in eq. (21) can not be done since $p\sim q$. We obtain that

$\begin{split}C_{1}(q)=g^{2}C_{F}T\rho^{\prime}_{00}(0,\mathbf{q})\,,\\\
C_{3}(q)=-\frac{g^{2}C_{F}\rho^{\prime}_{00}(0,\mathbf{q})}{4}\,,\\\
C_{2}(q)=0\,.\end{split}$ (65)

Actually, the last equation seems to be true at any perturbative order just
because of the structure of NRQCD. We can see explicitly that the conditions
in eqs. (60) and (63) are fulfilled.

Finally, let us discuss the physical picture that the EFT treatment of the
cases $p\sim T$ and $p\sim\sqrt{MT}$ provides when they are put together. If
we start with a heavy quark at rest with the medium, it will start to gain
momentum due to collisions. At this stage, the evolution of the system is
given by a Boltzmann equation like eq. (64). This evolution is such that there
is almost an equal probability to gain momentum than to lose it. Therefore, we
will naturally arrive to a situation in which $p\gg T$ after some time. In
this case, the evolution can already be described by a Langevin equation. As
the momentum keeps increasing, the drag force starts to become important.
Finally, we arrive to an equilibrium distribution in which momenta much larger
than $\sqrt{MT}$ are very rare. Let us note that the arguments needed to
arrive to these qualitative statements are valid even if the medium is
strongly coupled.

##  VII Discussion and conclusions

In this manuscript, we have discussed an EFT approach to the derivation of the
evolution of a heavy quark in a medium. Our main motivation was to pave the
way for future developments in the study of quarkonium suppression. We would
like to obtain a Lindblad equation for quarkonium without assuming a weakly-
coupled plasma and using the hierarchy of energy scales that appears in the
problem. This has already been done in the case that the medium sees
quarkonium as a small color dipole [10, 11]. However, we would like to
generalize to the case $Tr\sim 1$ as this would allow modeling more
realistically excited states of bottomonium and charmonium at LHC energies.
This is challenging because the EFT in which heavy quarkonium is best
described is pNRQCD, and it is obtained after integrating out the scale $1/r$.
However, if $Tr\sim 1$ the matching between NRQCD and pNRQCD is modified by
the medium. The appearance of an imaginary part of the potential indicates
that the medium induces dissipative effects that can not be encoded in an
effective action, instead we need an influence functional in which terms
mixing the two branches of the Schwinger-Keldysh contour appear. Therefore, we
need to modify pNRQCD to include this kind of terms. The motivation of this
manuscript is to study a simpler case in order to pave the way for this
modification of pNRQCD in the future.

The simpler case we studied is a single heavy quark interacting with a medium.
We did it following the EFT philosophy, in which each energy scale must be
treated separately using a series of EFTs. We distinguished two cases, the
case $p\sim\sqrt{MT}$ and the case $p\sim T$. In both cases we start with
NRQCD, however, in the first case we integrate out the scale $\sqrt{MT}$ as an
intermediate step going from NRQCD to $\textrm{NRQCD}_{sh}$. Finally, we
integrate the scale $T$ in both cases to go to a new EFT that we named LET.
The structure of this EFT is given by eq. (37), which is one of the main
results of this manuscript. This formula combines the constraints of the
Keldysh symmetry together with the special features of heavy particles in the
dilute limit. Further constraints are provided by the fluctuation-dissipation
theorem, which together with the EFT power counting leads to eq. (53) for
$p\sim\sqrt{MT}$ and to eq. (64) for $p\sim T$. This is the second main result
from this manuscript, that eqs. (53) and (64) can be obtained simply by
symmetry and power counting arguments within the EFT approach. The physical
picture is also quite transparent, a heavy quark with momentum of order $T$
can be described by a Boltzmann equation and will slowly increase its momentum
until we reach a moment in which the momentum is large enough so that a
Langevin equation becomes valid.

Let us now discuss in more details the relevance of the study made in this
manuscript in the context of quarkonium suppression. We have seen that the
state of a heavy quark can not be described by a single position $x$, instead
we need the position in the upper(lower) Schwinger-Keldysh branch $x_{1}$,
($x_{2}$). Note that $x_{1}-x_{2}$ is related to $p$ by the Wigner transform.
The increase of $p$ we discussed below is seen in coordinate space as a
narrowing of the $x_{1}-x_{2}$ coordinate or, put differently, the density
matrix becoming almost diagonal in the coordinate basis. This behavior has
also been observed in quarkonium [48, 49]. One main difference between the
case of quarkonium and that of a single heavy quark is that the number of
coordinates needed to describe the system doubles. In quarkonium, we can talk
about the center-of-mass coordinate, $R$, and the relative coordinate, $r$.
However, in non-equilibrium situations we need to consider the doubling of
degrees of freedom, and then we have to take into account four coordinates,
$R_{1}$, $R_{2}$, $r_{1}$ and $r_{2}$. A complete EFT treatment of quarkonium
would identify all the possible scale hierarchies that can be constructed with
these coordinates and that appear in the problem of quarkonium suppression.
The case studied in this manuscript corresponds to a situation in which the
quark and the antiquark are very far apart, and we have studied the cases
$x_{1}-x_{2}\sim\frac{x_{1}+x_{2}}{2}$ and
$x_{1}-x_{2}\ll\frac{x_{1}+x_{2}}{2}$. In [10, 11], the case studied
corresponds to $r_{1}-r_{2}\sim\frac{r_{1}+r_{2}}{2}\ll\frac{1}{T}$ and $T\gg
E$. Perturbative studies have shown that the Lindblad equation that appears in
the case $T\gg E$ naturally leads to a decrease of $r_{1}-r_{2}$. This
produces a change in the power counting that, among other things, makes the
drag force not a perturbation (NLO corrections in $E/T$ that include the drag
force were added in [50, 51]). Regarding the relation between the scales
$1/r$, $T$ and $E$; pNRQCD has been studied for all possible relations [7, 9,
8] but only taking into account the analogous to $S_{i}$ part in the notation
of eq. (22). This is enough to discuss spectroscopy, but not to study the
evolution of quarkonium inside a plasma. What is missing, and we hope this
work helps to develop, is the construction of the analogous to $\mathcal{I}$
in pNRQCD.

Finally, let us discuss possible extensions of the present work in the context
of heavy quark propagation itself. As in any EFT, an obvious improvement would
be to compute higher order corrections in the $\sqrt{\frac{T}{M}}$ expansion
to improve our knowledge about the evolution of $f(\mathbf{p})$. Another
possible direction is to relax the assumptions about the symmetries of the
problem. For example, until now we have assumed that both the medium and the
distribution of heavy quarks are homogeneous in space and isotropic. It would
be interesting to relax these conditions, as they are not completely fulfilled
in heavy-ion collisions. Finally, the case $p\gg\sqrt{MT}$ is also
interesting. In this case, we would be studying the case of a heavy quark that
loses energy until thermalizing with the medium.

###### Acknowledgements.

MAE wants to thanks the careful reading of a first version of this manuscript
and the useful comments of Juan-Torres Rincon, Joan Soto, Nora Brambilla and
Antonio Vairo. The work of MAE has been supported by the Maria de Maetzu
excellence program under project CEX2019-000918-M, by the Spanish Research
State Agency under project PID2019-105614GB-C21 and by the grant 2021-SGR-249
of Generalitat de Catalunya.

##  Appendix A Feynmann rules for $\textrm{NRQCD}_{sh}$

The Lagrangian of the heavy quark sector of $\textrm{NRQCD}_{sh}$ is

$\mathcal{L}_{\xi}=\sum_{\mathbf{p}\neq
0}\xi^{\dagger}_{\mathbf{p}}\\{iD_{0}-\frac{p^{2}}{2M}+i\frac{\mathbf{p}\cdot(\boldsymbol{\nabla}+ig\mathbf{A})}{M}+\frac{(\boldsymbol{\nabla}+ig\mathbf{A})^{2}}{2M}+c_{4}\frac{p^{4}}{8M^{3}}+c_{F}g\frac{\boldsymbol{\sigma}\cdot\mathbf{B}}{2M}\\}\xi_{\mathbf{p}}+\cdots$
(66)

now on we will not write explicit the sub-script $\mathbf{p}$ in this section
as this quantity is conserved in all diagrams. They Feynmann rules for this
theory are

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{}{}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@lineto{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{}}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{25.6073pt}{-2.84544pt}\pgfsys@moveto{25.6073pt}{-2.84544pt}\pgfsys@lineto{25.6073pt}{2.84502pt}\pgfsys@lineto{31.29776pt}{2.84502pt}\pgfsys@lineto{31.29776pt}{-2.84544pt}\pgfsys@closepath\pgfsys@moveto{31.29776pt}{2.84502pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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(73) $\leavevmode\hbox to57.31pt{\vbox
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\frac{ic_{4}p^{4}}{8M^{3}}\,,$
(74)

and finally

$\leavevmode\hbox to75.38pt{\vbox
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@moveto{31.45276pt}{0.0pt}\pgfsys@curveto{31.45276pt}{1.65688pt}{30.10963pt}{3.0pt}{28.45276pt}{3.0pt}\pgfsys@curveto{26.79588pt}{3.0pt}{25.45276pt}{1.65688pt}{25.45276pt}{0.0pt}\pgfsys@curveto{25.45276pt}{-1.65688pt}{26.79588pt}{-3.0pt}{28.45276pt}{-3.0pt}\pgfsys@curveto{30.10963pt}{-3.0pt}{31.45276pt}{-1.65688pt}{31.45276pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\frac{c_{F}g(\mathbf{q}\times\boldsymbol{\sigma})}{2M}\,.$
(75)

##  Appendix B Gauge invariance of $\mathcal{I}$

In this appendix, we are going to discuss the gauge invariance of eq. (26) in
more detail. Let us focus in the simplest term that is yet already problematic

$-i\Gamma\int\,d^{4}x\xi^{\dagger}_{2}\xi_{1}\,.$ (76)

As we discussed, this terms is only invariant under a restricted class of
gauge transformations, those that transform in the same way fields in the two
branches of the Schwinger-Keldysh contour. This term can be made gauge-
invariant by including two extra Wilson line.

$-i\Gamma\int\,d^{4}x\xi^{\dagger}_{2}U_{2}(\mathbf{x};t,\infty)U_{1}(\mathbf{x};\infty,t)\xi_{1}\,,$
(77)

where $U_{i}$ is a temporal Wilson line made of fields of type $i$. This term
is gauge-invariant because fields of type $1$ at $t\to\infty$ are identical to
fields of type $2$ at $t\to\infty$. This is a consequence of the largest time
equation [52] that implies that any correlator in which a quantum field (in
the sense of eq. (32)) has the largest time argument is zero. In other words,
quantum fields are zero at $t\to\infty$ and so are their gauge transformation.

Let us now discuss the differences with [20] and other words regarding an EFT
for hydrodynamics. There fields of type $1$ (or $2$) were assumed to appear
always in gauge-invariant conbinations thanks to the introduction of
Stueckelberg-like fields [53]. However, this solution is not suitable to our
case since the generalization of the Stueckelberg theory to non-Abelian
symmetries is problematic [53]. Note that in [20] they are dealing with a
theory for the field $A_{\mu}$ in which only derivatives of this field can
appear in the influence functional. This is not analog to our case in which
since the field $\xi$ does appear without derivates in most of the terms we
are interested.

Now we can discuss the issue if the fields in $D_{0}$ and $\mathbf{B}$
appearing in eq. (26) are of type $1$ or $2$. Let us first note that if we
impose that $S_{LET}=0$ in the limit $\xi_{Q}=A_{Q}=0$ then $\mathbf{B}$ and
the $A_{0}$ field inside $D_{0}$ need to be evaluated at time $t$. Regarding
$D_{0}$, there is only one possible combination because it can be checked that

$\xi^{\dagger}_{2}D_{0,2}U_{2}(\mathbf{x};t,\infty)U_{1}(\mathbf{x};\infty,t)\xi_{1}=\xi^{\dagger}_{2}U_{2}(\mathbf{x};t,\infty)U_{1}(\mathbf{x};\infty,t)D_{0,1}\xi_{1}\,,$
(78)

where $D_{0,i}=\partial_{0}-igA_{0,1}$. Regarding $B$, it can enter in the
following combinations

$\begin{split}\xi^{\dagger}_{2}B_{2}U_{2}(\mathbf{x};t,\infty)U_{1}(\mathbf{x};\infty,t)\xi_{1}\,,\\\
\xi^{\dagger}_{2}U_{2}(\mathbf{x};t,\infty)U_{1}(\mathbf{x};\infty,t)B_{1}\xi_{1}\,.\end{split}$
(79)

Therefore, to be more precised, in eq. (26) we should change

$\int\,d^{4}x\xi^{\dagger}_{2}\frac{\boldsymbol{\sigma}\cdot\mathbf{B}}{M}\xi_{1}\,,$
(80)

into

$\begin{split}\frac{1}{2}\int\,d^{4}x\xi^{\dagger}_{2}\frac{\boldsymbol{\sigma}\cdot\mathbf{B_{2}}}{M}U_{2}(\mathbf{x};t,\infty)U_{1}(\mathbf{x};\infty,t)\xi_{1}\\\
+\frac{1}{2}\int\,d^{4}x\xi^{\dagger}_{2}U_{2}(\mathbf{x};t,\infty)U_{1}(\mathbf{x};\infty,t)\frac{\boldsymbol{\sigma}\cdot\mathbf{B_{1}}}{M}\xi_{1}\,.\end{split}$
(81)

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