# Schwinger-Keldysh effective action for relativistic Brownian particle in
AdS-CFT

Yanyan Bu<EMAIL_ADDRESS>Biye Zhang<EMAIL_ADDRESS>(correspondence
author)

###### Abstract

We compute Schwinger-Keldysh effective action for a relativistic heavy quark
(with constant background velocity) in strongly coupled $\mathcal{N}=4$
supersymmetric Yang-Mills plasma. The holographic dual description involves a
noisy trailing string moving in Schwarzschild-AdS5 black brane. The noise is
caused by Hawking radiation emitted from string worldsheet horizon. Besides
quadractic terms, the effective action contains cubic interactions, which are
entirely induced by the constant background velocity. The nonlinear
fluctuation-dissipation relations for three-point functions are discussed
based on holographic results.

###### Contents

  1. 1 Introduction
  2. 2 Holographic setup
    1. 2.1 General statement
    2. 2.2 Trailing string in Schwarzschild-AdS5 and holographic SK contour
    3. 2.3 String perturbation
  3. 3 Solution for string fluctuation on the contour
    1. 3.1 Horizon gluing conditions
    2. 3.2 From linearly independent solutions to solution on the contour
  4. 4 Schwinger-Keldysh effective action for Brownian particle
    1. 4.1 Quadratic action: coloured noise
    2. 4.2 Cubic action: non-Gaussianity
  5. 5 Summary and Outlook
  6. A Perturbative $S_{\rm NG}$ in Schwarzschild coordinate
  7. B Subtlety due to non-commutativity of $\epsilon\to 0$ versus $\hat{\omega}\to 0$
  8. C KMS conditions for three-point functions

##  1 Introduction

Brownian motion of a heavy particle in a thermal bath has become a textbook
example [1, 2] for understanding fundamental aspects of non-equilibrium
statistical mechanics. Langevin equation is perhaps the simplest effective
treatment over complicated dynamics of Brownian motion:

$\displaystyle M_{\rm
kin}\frac{d^{2}}{dt^{2}}q(t)+\eta\frac{d}{dt}q(t)=\xi(t),$ (1.1)

where $q(t)$ is position of Brownian particle, $\eta$ represents the drag
force, and the stochastic variable $\xi(t)$ stands for random force felt by
the Brownian particle. Usually, the random force is assumed to be white noise:

$\displaystyle\langle\xi(t)\rangle=0,\qquad\langle\xi(t)\xi(t^{\prime})\rangle=2T\eta\delta(t-t^{\prime}),$
(1.2)

where the strength of noise-noise correlator is related to drag force through
fluctuation-dissipation theorem. Here, $T$ is the temperature of thermal bath.

The Langevin theory of Brownian motion (1.1) could be derived from real time
path integral on closed time contour [3, 4] (see also [5, 6]). The latter is
also called Schwinger-Keldysh (SK) formalism [7, 8, 9], which provides a
unified framework for describing quantum many-body system in and out of
equilibrium [9]. Within the SK formalism, the quantum system effectively
evolves forward (from initial time $t_{i}$ to final time $t_{f}$) and then
backward (from $t_{f}$ to $t_{i}$), forming a closed time contour, as depicted
in Figure 1.

$U(t_{f},t_{i})$$t_{f}$$U^{\dagger}(t_{f},t_{i})$$\rho_{0}$$t_{i}$ Figure 1:
The SK closed time contour: $\rho_{0}$ is the initial density matrix, and
$U(t_{f},t_{i})$ is the time-evolution operator from initial time $t_{i}$ to
final time $t_{f}$.

Moreover, the SK formalism systematically incorporates both fluctuations and
dissipations, which is achieved by the doubling of degrees of freedom:
$q\to(q_{1},q_{2})$, where the subscripts $1,2$ denote variables on the upper
and lower branches of the SK contour in Figure 1.

Brownian motion involves two key concepts in non-equilibrium statistical
mechanics: closed system versus open system [1]. Thinking of the Brownian
particle and the thermal bath as a whole forms a closed system, denoted as
$\\{q,\Phi\\}$. Dynamics of this closed system is presumably described by an
action:

$\displaystyle S_{C}=S_{p}[q]+S_{th}[\Phi]+S_{int}[q,\Phi],$ (1.3)

where $S_{p}[q]$ is action for the Brownian particle, $S_{th}[\Phi]$ describes
the microscopic theory of the constitutes (collectively denoted as $\Phi$) for
the thermal bath, and $S_{int}[q,\Phi]$ is responsible for the interactions
between the Brownian particle and constituents of the thermal bath. The
partition function for the closed system is

$\displaystyle Z=\int[Dq][D\Phi]e^{iS_{C}}.$ (1.4)

However, the microscopic theory (1.3) contains too many degrees of freedom and
is very hard (if not impossible) to study in general. On the other hand, the
main interest is on the behavior of the Brownian particle. This motivates the
concept of open system: here, it is the Brownian particle itself, denoted as
$\\{q\\}$. Moreover, this open system will be described by some effective
theory, in which the only degree of freedom is the Brownian particle. In
principle, the desired effective theory could be obtained by integrating out,
in (1.4), all the degrees of freedom for the thermal bath. Then, the partition
function (1.4) is equivalently expressed as a path integral over position of
the Brownian particle only,

$\displaystyle Z=\int[Dq]e^{iS_{eff}[q]}.$ (1.5)

Certainly, in order to systematically incorporate fluctuations and
dissipations, such a Wilsonian renormalization group (RG) flow should be
carried out over the SK closed time contour. The resultant SK effective action
$S_{eff}[q]$ will encode information of the thermal bath, interactions between
the Brownian particle and the thermal bath, as well as the state.

In practice, the “integrating out” procedure outlined above is very
challenging to carry out111In simple examples, the degrees of freedom for the
thermal bath could be integrated out, giving linear Langevin equation [3] and
its nonlinear extensions [10, 11]. In these examples, microscopic details of
the thermal bath are simplified as a collection of harmonic oscillators.
Moreover, the interaction between the Brownian particle and bath oscillator
was assumed to be either Gaussian [3] or weakly cubic [10, 11].. It is thus
natural to turn to effective field theory (EFT) approach, and construct the
effective action based on symmetry principle. In recent years, in order to
tackle with dissipation, the SK formalism has been applied to formulate an EFT
for dissipative hydrodynamics [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24, 25] (see [26] for a pedagogical review). This significant achievement for
hydrodynamic EFT has been largely inspired by the AdS-CFT correspondence [27,
28, 29], which provides a tractable framework for studying dynamics of
strongly coupled large $N_{c}$ (the number of colors) gauge theory via weakly
coupled gravitational theory in asymptotic AdS space.

Brownian motion of heavy quark became an important example in revealing
structure of thermal noise in AdS-CFT. A heavy quark in AdS-CFT [30, 31, 32,
33, 34] is represented as an open string stretching from the horizon up to a
probe brane, and the string’s endpoint on the probe brane is quark’s
position222Interestingly, the quark string model was recently taken to
understand chaos and realize holographic dual of SYK model [35, 36, 37].. The
stochastic Langevin equation (1.1) was derived in [5] (see also [38]) via
Krustal extension [39] of finite temperature AdS-CFT [40], in which
fluctuation-dissipation theorem turns out to be an outcome of the
correspondence, rather than being imposed by hand [40]. This study was further
extended in a number of aspects: the heavy quark could be relativistic [41,
42], the environment would be in a non-equilibrium state corresponding to
time-dependent AdS black hole [43], the dissipation (the noise) would be
nonlinear (non-Gaussian) [44], etc. Particularly, the latter adopted the
holographic SK contour [45] (see [46, 47, 48] for an alternative holographic
prescription for SK contour) and derived the effective action for a nonlinear
Langevin theory [10, 11]. Over the past two years, the holographic SK contour
[45] attracted a lot of attention in the study of SK field theory [44, 49, 50,
51, 52, 53, 54, 55, 56].

In this work, we will employ the proposal [45] and explore leading order
nonlinear corrections to relativistic Langevin theory of [41, 42]. A heavy
quark in a steady state motion (i.e., with a constant background velocity)
corresponds to a trailing string solution in the target AdS black brane [30,
42]. The trailing string solution induces an event horizon on the worldsheet,
which is different from that of target space. As clarified in [41, 42], it is
the Hawking radiation behind this worldsheet horizon that causes fluctuations
on the heavy quark. Thus, the emergent AdS black brane on the worldsheet will
be doubled in the spirit of [45]. Perturbing around the trailing string
background, we expand the string’s Nambu-Goto action in amplitude of string
perturbation: besides quadratic terms, the action also contains cubic
interactions. The cubic terms are entirely induced by quark’s background
velocity, and are not present in [44]. Intriguingly, these cubic terms
explicitly break worldsheet time-reversal ($\mathcal{T}$) symmetry. The cubic
interactions generate nonlinear corrections to relativistic Langevin theory of
[41, 42], which do not satisfy Kubo-Martin-Schwinger (KMS) conditions for
three-point functions of medium force [57, 17]. This supports the claim that
the fluctuations and dissipations caused by the worldsheet horizon shall be of
non-thermal nature [41].

The rest of this paper will be structured as follows. In section 2, we present
the relativistic trailing string in AdS5 black brane, which is the holographic
dual of a relativistic heavy quark in $\mathcal{N}=4$ SYM plasma. We also
analyze nonlinear dynamics for classical string perturbation around the
trailing string background. In section 3, we analyze the solution for
classical string perturbation on the holographic SK contour associated with
the string worldsheet horizon. In section 4, we compute the SK effective
action for the relativistic heavy quark, up to cubic order in the quark’s
position. At quadratic order, we calculate the SK effective action first
analytically in the hydrodynamic limit and then numerically for arbitrary
value of frequency. At cubic order, we compute the SK effective action
analytically in small frequency regime. In section 5, we make a summary of
this work and outlook future directions. Appendices A, B and C supplement
further calculational details.

##  2 Holographic setup

###  2.1 General statement

The thermal bath is holographically described by Schwarzschild-AdS5 black
brane (the AdS radius has been set to unity):

$\displaystyle ds^{2}_{\rm
TG}=g_{MN}dX^{M}dX^{N}=\frac{1}{z^{2}}\left[\frac{dz^{2}}{f(z)}-f(z)dt^{2}+\delta_{ij}dX^{i}dX^{j}\right],\qquad
i,j=1,2,3,$ (2.1)

where $f(z)=1-z^{4}/z_{h}^{4}$. The Hawking temperature is $T=1/(\pi z_{h})$,
which is also the temperature of thermal bath. According to AdS/CFT
correspondence, dynamics of a heavy quark is determined by an open string
moving in the target space (2.1), see [30, 32, 31]. The open string stretches
from the event horizon $z=z_{h}$ up to a probe brane placed near the AdS
boundary, with string’s endpoint on the probe brane identified with position
of the heavy quark. In this work, we will focus on an infinitely heavy quark,
which corresponds to placing the probe brane at the AdS boundary $z=0$.

Dynamics of an open string is described by Nambu-Goto action

$\displaystyle S_{\rm NG}=-\frac{1}{2\pi\alpha^{\prime}}\int
d^{2}\sigma\sqrt{-h(X)},$ (2.2)

where $h$ is determinant of the induced metric $h_{ab}$ on string worldsheet:

$\displaystyle ds^{2}_{\rm
WS}=h_{ab}d\sigma^{a}\sigma^{b}=g_{MN}\frac{\partial
X^{M}}{\partial\sigma^{a}}\frac{\partial X^{N}}{\partial\sigma^{b}}.$ (2.3)

In static gauge, the worldsheet coordinate is
$\sigma^{a}\equiv(\sigma,\tau)=(z,t)$, and embedding of the open string in the
target space (2.1) is specified by the spatial coordinates
$X^{i}(\sigma^{a})$.

In presence of an external Maxwell field, the Nambu-Goto action (2.2) should
be supplemented by a boundary term:

$\displaystyle S_{\rm bdy}=\int d\tau
A_{M}(X)\frac{dX^{M}}{d\tau}\bigg{|}_{\rm bdy}=-\int d\tau
X^{M}\partial_{\tau}A_{M}(X)\big{|}_{\rm bdy},$ (2.4)

where in the second equality we ignored a total time-derivative term.

The partition function for the bulk theory is

$\displaystyle Z_{\rm bulk}=\int[DX][Dg_{MN}]e^{iS_{\rm bulk}[X,\,g_{MN}]},$
(2.5)

where $S_{\rm bulk}$ is the total action for the bulk theory. In probe limit,
the target space (2.1) does not fluctuate. Then, the bulk partition function
$Z_{\rm bulk}$ gets reduced into that of an open string in the curved
background spacetime (2.1):

$\displaystyle Z_{\rm bulk}\simeq Z_{\rm string}=\int[DX]e^{iS[X]},$ (2.6)

where $S$ is the total string action

$\displaystyle S=S_{\rm NG}+S_{\rm bdy}.$ (2.7)

It will be clear that the string embedding profile $X$ is a functional of
quark’s position $q$, i.e., $X=X[q]$. Thus, the bulk path integral (2.6) will
be eventually cast into a path integral over the position $q$. We will work in
the saddle point approximation:

$\displaystyle Z_{\rm string}=\int[Dq]e^{iS[X[q]]},$ (2.8)

where $S[X[q]]$ is the on-shell classical string action. The AdS-CFT
conjectures that $Z$ of (1.4) is equivalent to $Z_{\rm bulk}$. Thus, in the
probe limit, the on-shell string action $S[X[q]]$ will be identified with the
effective action $S_{eff}[q]$ for Brownian particle in plasma medium.
Therefore, derivation of $S_{eff}[q]$ boils down to solving the classical
equation of motion (EOM) for an open string in Schwarzschild AdS5.

###  2.2 Trailing string in Schwarzschild-AdS5 and holographic SK contour

We are interesting in steady state motion of a heavy quark in $\mathcal{N}=4$
SYM plasma. On the gravity side, this corresponds to a trailing string in
Schwarzschild-AdS5 [30, 32]. Consider a “steady” profile for the string
embedding in the target space:

$\displaystyle\bar{X}^{1}=\beta
t+\beta\zeta(z),\qquad\bar{X}^{2}=\bar{X}^{3}=0,$ (2.9)

where we use “bar” to denote the steady state configuration. The trailing
string solution was presented in [30, 32]:

$\displaystyle\zeta(z)=\frac{1}{2\pi T}\left[\tan^{-1}(z\pi T)-\tanh^{-1}(z\pi
T)\right].$ (2.10)

For the configuration (2.10), the energy and momentum will flow down the
string into the horizon of target space. Therefore, such a trailing string
solution has to be maintained by a constant external electric field [30]. The
relation between the velocity $\beta$ and electric field $E$ is determined by
the demand that the string’s action is time independent and real [42]:

$\displaystyle\beta=\frac{2E\alpha^{\prime}}{\sqrt{\pi^{2}T^{4}+4E^{2}\alpha^{\prime
2}}}=\frac{2E}{\sqrt{\lambda_{\rm tH}}\pi
T^{2}}\frac{1}{\sqrt{1+\frac{4E^{2}}{\lambda_{\rm tH}\pi^{2}T^{4}}}},$ (2.11)

where $\lambda_{\rm tH}=1/\alpha^{\prime 2}$ is the ’t Hooft coupling constant
[27].

Plugging the trailing string solution (2.9) and (2.10) into (2.1), we obtain
induced metric on the worldsheet:

$\displaystyle d\bar{s}^{2}_{\rm
WS}=\frac{1}{z^{2}}\frac{f(z)+\beta^{2}z^{4}/z_{h}^{4}}{f^{2}(z)}dz^{2}-\frac{2\beta^{2}}{z_{h}^{2}f(z)}dzdt-\frac{1}{z^{2}}[f(z)-\beta^{2}]dt^{2}.$
(2.12)

Apparently, an event horizon is developed on the worldsheet at
$z=z_{h}(1-\beta^{2})^{1/4}$, which is different from event horizon of target
space (2.1) once $\beta\neq 0$. In order to diagonalize the worldsheet metric
(2.12), we turn to the “hatted” coordinate system [42]:

$\displaystyle\hat{t}=\frac{t+\zeta_{1}(z)}{\sqrt{\gamma}},\qquad\qquad\hat{z}=\sqrt{\gamma}z,\qquad\qquad\hat{X}^{i}=\sqrt{\gamma}X^{i},$
$\displaystyle\zeta_{1}(z)\equiv\frac{1}{2\pi T}\left[\tan^{-1}(z\pi
T)-\tanh^{-1}(z\pi T)\right]$
$\displaystyle\qquad\quad-\frac{\sqrt{\gamma}}{2\pi
T}\left[\tan^{-1}(z\sqrt{\gamma}\pi
T)-\frac{1}{2}\log\frac{1+z\sqrt{\gamma}\pi T}{1-z\sqrt{\gamma}\pi T}\right],$
(2.13)

which brings the worldsheet metric (2.12) to AdS type:

$\displaystyle d\bar{s}_{\rm
WS}^{2}=\frac{1}{{\hat{z}}^{2}}\left[\frac{d\hat{z}^{2}}{f(\hat{z})}-f(\hat{z})d\hat{t}^{2}\right].$
(2.14)

Here, the Lorentz boost factor $\gamma=1/\sqrt{1-\beta^{2}}$.

In the ingoing EF coordinate system $\sigma^{a}=(\hat{z},\hat{v})$ with
$d\hat{v}=d\hat{t}-d\hat{z}/f(\hat{z})$, the worldsheet metric is free of
coordinate singularity:

$\displaystyle d\bar{s}^{2}_{\rm
WS}=\frac{1}{\hat{z}^{2}}\left[-2d\hat{v}d\hat{z}-f(\hat{z})d\hat{v}^{2}\right].$
(2.15)

We will be interesting in the fluctuations caused by Hawking radiation emitted
from the worldsheet event horizon. Under the philosophy of [45], we double the
worldsheet AdS, and analytically continue it around the emergent event horizon
$\hat{z}=z_{h}$. Resultantly, the radial coordinate $\hat{z}$ varies along the
contour of Figure 2. This holographic contour forms an ideal prescription for
studying effective action for a non-equilibrium problem.

$0_{1}$$0_{2}$$z_{h}$Re$(\hat{z})$Im$(\hat{z})$$0_{1}$$0_{2}$$\hat{z}=z_{h}$$\epsilon$
Figure 2: From complexified (analytically continued near horizon) double AdS
(left) [58] to the holographic SK contour (right) [45]. Indeed, the two
horizontal legs overlap with the real axis.

The relationship between $\hat{t}$ and $\hat{v}$ is summarized below,

$\displaystyle\hat{v}$
$\displaystyle=\hat{t}+\chi_{2}(\hat{z}),\qquad\qquad\chi_{2}(\hat{z})\equiv-\int_{0_{2}}^{\hat{z}}\frac{dy}{f(y)},\qquad\hat{z}\in[0_{2},z_{h}+\epsilon],$
$\displaystyle\hat{v}$
$\displaystyle=\hat{t}+\chi_{1}(\hat{z}),\qquad\qquad\chi_{1}(\hat{z})\equiv-\int_{0_{1}}^{\hat{z}}\frac{dy}{f(y)},\qquad\hat{z}\in[0_{1},z_{h}+\epsilon],$
(2.16)

where the integration constants are fixed so that $\hat{v}$ and $\hat{t}$
equal on both AdS boundaries.

###  2.3 String perturbation

In order to study stochastic motion of the heavy quark, we turn on string
perturbation on top of the trailing string background reviewed in subsection
2.2. Without loss of generality, we turn on string perturbation in the
following way,

$\displaystyle\hat{X}^{1}(\hat{z},\hat{v})=\sqrt{\gamma}\beta\left[\sqrt{\gamma}\hat{t}(\hat{v})-\zeta_{1}(\hat{z})+\zeta(\hat{z})\right]+\hat{X}_{L}(\hat{z},\hat{v}),$
$\displaystyle\hat{X}^{2}(\hat{z},\hat{v})=0+\hat{X}_{T}(\hat{z},\hat{v}),\qquad\qquad\hat{X}^{3}(\hat{z},\hat{v})=0,$
(2.17)

where we only need the relation $\hat{t}(\hat{v})$ in its differential form.
Here, the string perturbations $\hat{X}_{L}$ and $\hat{X}_{T}$ are small in
comparison with their backgrounds, so that the Nambu-Goto action (2.2) could
be expanded in powers of $\hat{X}_{L}$ and $\hat{X}_{T}$:

$\displaystyle S_{\rm NG}=-\frac{1}{2\pi\alpha^{\prime}}\int
d\hat{v}\int_{0_{1}}^{0_{2}}d\hat{z}\sqrt{-\hat{h}(\hat{X})}=S_{\rm
NG}^{(0)}+S_{\rm NG}^{(1)}+S_{\rm NG}^{(2)}+S_{\rm NG}^{(3)}+\cdots,$ (2.18)

where the radial integral is along the $\hat{z}$-contour of Figure 2 in a
counter-clockwise way. Note that $\hat{X}^{i}(\hat{z}\to 0)/\sqrt{\gamma}$ is
the position of string endpoint. This shall be reflected in the AdS boundary
conditions for $X_{T},X_{L}$ to be presented later.

The zeroth order action vanishes since the integrand does not have any
singularity inside the radial contour:

$\displaystyle S_{\rm NG}^{(0)}=-\frac{1}{2\pi\alpha^{\prime}}\int
d\hat{v}\int_{0_{1}}^{0_{2}}\frac{d\hat{z}}{\hat{z}^{2}}=0.$ (2.19)

The next two terms in the expansion (2.18), say $S_{\rm NG}^{(1)}$ and $S_{\rm
NG}^{(2)}$, were previously discussed in [42] in the
$(\hat{z},\hat{t})$-coordinate system. Here, for completeness we record them
in the hatted ingoing EF coordinate system. The first order action $S_{\rm
NG}^{(1)}$ is actually a boundary term:

$\displaystyle S_{\rm NG}^{(1)}=$
$\displaystyle\frac{1}{2\pi\alpha^{\prime}}\int
d\hat{v}\int_{0_{1}}^{0_{2}}d\hat{z}\,\beta\gamma(\pi
T)^{2}\partial_{\hat{z}}\hat{X}_{L}$ $\displaystyle=$ $\displaystyle\int
d\hat{v}\int_{0_{1}}^{0_{2}}d\hat{z}\,E\partial_{\hat{z}}\hat{X}_{L}=\int
d\hat{v}E\hat{X}_{L}\bigg{|}_{0_{1}}^{0_{2}},$ (2.20)

which is exactly cancelled by the first order counterpart of boundary action
$S_{\rm bdy}$ (2.4). Here, we made use of the relation (2.11).

The quadratic order action $S_{\rm NG}^{(2)}$ is

$\displaystyle S_{\rm NG}^{(2)}=$
$\displaystyle-\frac{1}{2\pi\alpha^{\prime}}\int{d\hat{v}}\int_{0_{1}}^{0_{2}}{d\hat{z}}\frac{1}{\hat{z}^{2}}\left[-\gamma^{2}\partial_{\hat{v}}\hat{X}_{L}\partial_{\hat{z}}\hat{X}_{L}+\frac{\gamma^{2}}{2}f(\hat{z})(\partial_{\hat{z}}\hat{X}_{L})^{2}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\left.-\partial_{\hat{v}}\hat{X}_{T}\partial_{\hat{z}}\hat{X}_{T}+\frac{1}{2}f(\hat{z})(\partial_{\hat{z}}\hat{X}_{T})^{2}\right].$
(2.21)

Apparently, the transverse dynamics behaves the same as the static case [5],
while the longitudinal dynamics gets contracted by the Lorentz boost factor
$\gamma$.

The third order action $S_{\rm NG}^{(3)}$ will be the focus of present work,

$\displaystyle S_{\rm NG}^{(3)}=$
$\displaystyle-\frac{\beta\gamma}{2\pi\alpha^{\prime}}\int{d\hat{v}}\int_{0_{1}}^{0_{2}}d\hat{z}\left\\{\frac{\gamma^{2}\pi^{2}T^{2}}{2}f(\hat{z})(\partial_{\hat{z}}\hat{X}_{L})^{3}+\frac{\pi^{2}T^{2}}{2}f(\hat{z})\partial_{\hat{z}}\hat{X}_{L}(\partial_{\hat{z}}\hat{X}_{T})^{2}\right.$
$\displaystyle\left.-\frac{\gamma^{2}}{2\hat{z}^{2}}(-1+3\pi^{2}T^{2}\hat{z}^{2})(\partial_{\hat{z}}\hat{X}_{L})^{2}\partial_{\hat{v}}\hat{X}_{L}-\frac{1}{2\hat{z}^{2}}(1+\pi^{2}T^{2}\hat{z}^{2})(\partial_{\hat{z}}\hat{X}_{T})^{2}\partial_{\hat{v}}\hat{X}_{L}\right.$
$\displaystyle\left.+\frac{f(\hat{z})}{\hat{z}^{2}(1+\pi^{2}T^{2}\hat{z}^{2})}\partial_{\hat{z}}\hat{X}_{T}\partial_{\hat{z}}\hat{X}_{L}\partial_{\hat{v}}\hat{X}_{T}-\frac{\gamma^{2}}{\hat{z}^{2}[1+(\hat{z}\pi
T)^{2}]}\partial_{\hat{z}}\hat{X}_{L}(\partial_{\hat{v}}\hat{X}_{L})^{2}\right.$
$\displaystyle\left.-\frac{1}{\hat{z}^{2}(1+\pi^{2}T^{2}\hat{z}^{2})}\partial_{\hat{z}}\hat{X}_{L}(\partial_{\hat{v}}\hat{X}_{T})^{2}\right\\}.$
(2.22)

In order to analyze time-reversal symmetry for the bulk theory, we turn to
Schwarzschild coordinate system $\sigma^{a}=(\hat{z},\hat{t})$ momentarily, in
which the Nambu-Goto action $S_{\rm NG}^{(2)}$ and $S_{\rm NG}^{(3)}$ are
summarized in appendix A. From the worldsheet perspective, the second order
action $S_{\rm NG}^{(2)}$ is invariant under $\mathcal{T}$-transformation:
$\hat{t}\to-\hat{t},~{}\hat{X}_{L,T}(\hat{t},\hat{z})\to\hat{X}_{L,T}(-\hat{t},\hat{z})$.
However, the cubic order action $S_{\rm NG}^{(3)}$ explicitly breaks
$\mathcal{T}$-symmetry. $\mathcal{T}$-symmetry is crucial in deriving KMS
constraints on the SK effective action [17], which will be discussed in
section 4.

The EOMs for $\hat{X}_{T,L}$, derived from (2.21) and (2.22), are

$\displaystyle
0=\partial_{\hat{z}}\left[\frac{1}{\hat{z}^{2}}f(\hat{z})\partial_{\hat{z}}\hat{X}_{T}\right]-\frac{2}{\hat{z}^{2}}\partial_{\hat{v}}\partial_{\hat{z}}\hat{X}_{T}+\frac{2}{\hat{z}^{3}}\partial_{\hat{v}}\hat{X}_{T}+f_{T}[\hat{X}_{T},\hat{X}_{L}],$
$\displaystyle
0=\partial_{\hat{z}}\left[\frac{1}{\hat{z}^{2}}f(\hat{z})\partial_{\hat{z}}\hat{X}_{L}\right]-\frac{2}{\hat{z}^{2}}\partial_{\hat{v}}\partial_{\hat{z}}\hat{X}_{L}+\frac{2}{\hat{z}^{3}}\partial_{\hat{v}}\hat{X}_{L}+f_{L}[\hat{X}_{T},\hat{X}_{L}],$
(2.23)

where the nonlinear terms $f_{T},f_{L}$ are quadratic in
$\hat{X}_{T},\hat{X}_{L}$. We do not record explicit expressions for
$f_{T},f_{L}$ since they are irrelevant for the computation of cubic order SK
effective action. We will elaborate on this point later. The nonlinear EOMs
(2.23) will be solved perturbatively

$\displaystyle\hat{X}_{T,L}=\alpha^{1}\hat{X}_{T,L}^{(1)}+\alpha^{2}\hat{X}_{T,L}^{(2)}+\cdots.$
(2.24)

Then, at each order in the perturbative expansion (2.24),
$\hat{X}_{T,L}^{(n)}$ obeys linear differential equations. Particularly, at
the lowest order $\mathcal{O}(\alpha^{1})$, $\hat{X}_{T,L}$ satisfy the same
homogeneous equation:

$\displaystyle\partial_{\hat{z}}\left[\frac{1}{\hat{z}^{2}}f(\hat{z})\partial_{\hat{z}}\Phi\right]-\frac{2}{\hat{z}^{2}}\partial_{\hat{v}}\partial_{\hat{z}}\Phi+\frac{2}{\hat{z}^{3}}\partial_{\hat{v}}\Phi=0,\qquad\Phi=\hat{X}_{T}^{(1)}~{}~{}{\rm
or}~{}~{}\hat{X}_{L}^{(1)},$ (2.25)

which is analogous to EOM for a free scalar in Schwarzschild-AdS2. The AdS
boundary conditions are

$\displaystyle\hat{X}_{T}^{(1)}(\hat{z}\to
0_{s})=\hat{q}_{s}^{T},\qquad\qquad\hat{X}_{T}^{(n>1)}(\hat{z}\to
0_{s})=0,\qquad s=1,2,$ $\displaystyle\hat{X}_{L}^{(1)}(\hat{z}\to
0_{s})=\hat{q}_{s}^{L},\qquad\qquad\hat{X}_{L}^{(n>1)}(\hat{z}\to
0_{s})=0,\qquad s=1,2.$ (2.26)

Substituting the expansion (2.24) into (2.21), the quadratic order action
$S_{\rm NG}^{(2)}$ can be split into two parts:

$\displaystyle S_{\rm NG}^{(2)}=S_{\rm NG}^{(2),\rm P1}+S_{\rm NG}^{(2),\rm
P2},$ (2.27)

where

$\displaystyle S_{\rm NG}^{(2),\rm P1}=$
$\displaystyle-\frac{1}{2\pi\alpha^{\prime}}\int
d\hat{v}\int_{0_{1}}^{0_{2}}d\hat{z}\frac{1}{\hat{z}^{2}}\left[-\gamma^{2}\partial_{\hat{v}}\hat{X}^{(1)}_{L}\partial_{\hat{z}}\hat{X}^{(1)}_{L}+\frac{\gamma^{2}}{2}f(\hat{z})(\partial_{\hat{z}}\hat{X}^{(1)}_{L})^{2}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\left.-\partial_{\hat{v}}\hat{X}^{(1)}_{T}\partial_{\hat{z}}\hat{X}^{(1)}_{T}+\frac{1}{2}f(\hat{z})(\partial_{\hat{z}}\hat{X}^{(1)}_{T})^{2}\right],$
(2.28) $\displaystyle S_{\rm NG}^{(2),\rm P2}=$
$\displaystyle-\frac{1}{2\pi\alpha^{\prime}}\int
d\hat{v}\int_{0_{1}}^{0_{2}}d\hat{z}\frac{1}{\hat{z}^{2}}\left[-\partial_{\hat{v}}X_{T}^{(2)}\partial_{\hat{z}}\hat{X}_{T}^{(1)}-\partial_{\hat{v}}\hat{X}_{T}^{(1)}\partial_{\hat{z}}\hat{X}_{T}^{(2)}+f(\hat{z})\partial_{\hat{z}}\hat{X}_{T}^{(2)}\partial_{\hat{z}}\hat{X}_{T}^{(1)}\right.$
$\displaystyle\qquad\left.+\gamma^{2}\left(-\partial_{\hat{v}}\hat{X}_{L}^{(2)}\partial_{\hat{z}}\hat{X}_{L}^{(1)}-\partial_{\hat{v}}\hat{X}_{L}^{(1)}\partial_{\hat{z}}\hat{X}_{L}^{(2)}+f(\hat{z})\partial_{\hat{z}}\hat{X}_{L}^{(2)}\partial_{\hat{z}}\hat{X}_{L}^{(1)}\right)\right].$
(2.29)

In (2.27), a piece of order $\mathcal{O}(\alpha^{4})$ is ignored since it is
beyond scope of present work. Upon integration by parts, (2.29) is computed as

$\displaystyle S_{\rm NG}^{(2),\rm P2}=$
$\displaystyle-\frac{1}{2\pi\alpha^{\prime}}\int
d\hat{v}\int_{0_{1}}^{0_{2}}d\hat{z}\left\\{\partial_{\hat{z}}\left(\hat{X}_{T}^{(2)}\frac{f(\hat{z})}{\hat{z}^{2}}\partial_{\hat{z}}\hat{X}_{T}^{(1)}-\hat{X}_{T}^{(2)}\frac{1}{\hat{z}^{2}}\partial_{\hat{v}}\hat{X}_{T}^{(1)}\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.+\gamma^{2}\partial_{\hat{z}}\left(\hat{X}_{L}^{(2)}\frac{f(\hat{z})}{\hat{z}^{2}}\partial_{\hat{z}}\hat{X}_{L}^{(1)}-\hat{X}_{L}^{(2)}\frac{1}{\hat{z}^{2}}\partial_{\hat{v}}\hat{X}_{L}^{(1)}\right)\right\\},$
$\displaystyle=$ $\displaystyle-\frac{1}{2\pi\alpha^{\prime}}\int
d\hat{v}\left\\{\hat{X}_{T}^{(2)}\left(\frac{f(\hat{z})}{\hat{z}^{2}}\partial_{\hat{z}}\hat{X}_{T}^{(1)}-\frac{1}{\hat{z}^{2}}\partial_{\hat{v}}\hat{X}_{T}^{(1)}\right)\right.$
$\displaystyle\qquad\qquad\qquad\quad\left.+\gamma^{2}\hat{X}_{L}^{(2)}\left(\frac{f(\hat{z})}{\hat{z}^{2}}\partial_{\hat{z}}\hat{X}_{L}^{(1)}-\frac{1}{\hat{z}^{2}}\partial_{\hat{v}}\hat{X}_{L}^{(1)}\right)\right\\}\bigg{|}_{\hat{z}=0_{1}}^{\hat{z}=0_{2}}$
$\displaystyle=$ $\displaystyle 0,$ (2.30)

where the Dirichlet conditions for $\hat{X}_{T,L}^{(2)}$ at the AdS boundaries
have been utilized, cf. (2.26). Moreover, in obtaining (2.30), total time
derivative terms are ignored. In the same way, (2.28) is cast into a surface
term:

$\displaystyle S_{\rm NG}^{(2),\rm P1}=-\frac{1}{2\pi\alpha^{\prime}}\int
d\hat{v}\left[\frac{f(\hat{z})}{2\hat{z}^{2}}\hat{X}_{T}^{(1)}\partial_{\hat{z}}\hat{X}_{T}^{(1)}+\gamma^{2}\frac{f(\hat{z})}{2\hat{z}^{2}}\hat{X}_{L}^{(1)}\partial_{\hat{z}}\hat{X}_{L}^{(1)}\right]\bigg{|}_{\hat{z}=0_{1}}^{\hat{z}=0_{2}},$
(2.31)

where total time derivative terms are also ignored.

Therefore, accurate to cubic order $\mathcal{O}(\alpha^{3})$, the total string
action (Nambu-Goto action plus the boundary term) becomes

$\displaystyle S=S_{\rm NG}+S_{\rm bdy}=S_{\rm NG}^{(2),\rm P1}+S_{\rm
NG}^{(3)}\big{|}_{\hat{X}_{T,L}\to\hat{X}_{T,L}^{(1)}}.$ (2.32)

Thus, in order to derive boundary effective action up to cubic order in the
position $\hat{q}^{T,L}$, we just need to solve the linearized EOMs (2.25) for
string perturbations. This is similar to the case of a static heavy quark in
plasma medium [44]. Once the linearized EOMs (2.25) are solved, the core task
will be to compute the cubic action $S_{\rm
NG}^{(3)}\big{|}_{\hat{X}_{T,L}\to\hat{X}_{T,L}^{(1)}}$, which involves a
number of radial contour integrals.

##  3 Solution for string fluctuation on the contour

In this section, we solve the linearized EOMs (2.25) over the radial contour
of Figure 2. The solutions were previously obtained in [44] through heuristic
analysis with the help of time-reversal symmetry. Here, we elaborate on the
derivation for completeness, following the idea of [54]. To this end, we cut
the contour in Figure 2 at $\hat{z}=\hat{z}_{\pm}\equiv\hat{z}_{h}+\epsilon$,
where the subscripts “+” and “-” are used to distinguish between the upper and
lower segments of the contour. Then, we will search for general solutions for
(2.25) in the single copy of the contour, namely, either on the upper branch
or on the lower branch of Figure 2. Crucially, the general solution for the
upper branch and that for the lower branch have to be carefully glued at the
cutting point $\hat{z}=\hat{z}_{h}+\epsilon$. The correct gluing conditions at
the cutting point $\hat{z}=\hat{z}_{h}+\epsilon$ will be derived from the
quadratic order action (2.21).

###  3.1 Horizon gluing conditions

Since the actions for $\hat{X}_{T}^{(1)}$ and $\hat{X}_{L}^{(1)}$ differ by an
overall constant $\gamma^{2}$, the horizon gluing conditions for
$\hat{X}_{T}^{(1)}$ and $\hat{X}_{L}^{(1)}$ shall be the same. So, we will
take $\hat{X}_{T}^{(1)}$ as an example to derive the horizon gluing
conditions. The action for $\hat{X}_{T}^{(1)}$ is

$\displaystyle S_{\rm NG}^{(2),\rm
T}=-\frac{1}{2\pi\alpha^{\prime}}\int{d\hat{v}}\int_{0_{1}}^{0_{2}}{d\hat{z}}\frac{1}{\hat{z}^{2}}\left[-\partial_{\hat{v}}\hat{X}_{T}^{(1)}\partial_{\hat{z}}\hat{X}^{(1)}_{T}+\frac{1}{2}f(\hat{z})\left(\partial_{\hat{z}}\hat{X}^{(1)}_{T}\right)^{2}\right].$
(3.1)

In the spirit of [47], the gluing conditions at the cutting point can be
derived by demanding the bulk action $S_{\rm NG}^{(2),\rm T}$ to be extremal
with respect to variation of horizon data
$\hat{X}_{T}^{(1)}(\hat{z}_{h}+\epsilon,\hat{v})$. The variation of (3.1) is

$\displaystyle\delta S_{\rm NG}^{(2),\rm T}=$
$\displaystyle-\frac{1}{2\pi\alpha^{\prime}}\int
d\hat{v}\left\\{\left(-\frac{1}{\hat{z}^{2}}\partial_{\hat{v}}\hat{X}_{T}^{(1)}+\frac{f(\hat{z})}{\hat{z}^{2}}\partial_{\hat{z}}\hat{X}_{T}^{(1)}\right)\delta\hat{X}_{T}^{(1)}\bigg{|}_{0_{1}}^{\hat{z}_{-}}\right.$
$\displaystyle\qquad\qquad\qquad\quad\left.-\left(-\frac{1}{\hat{z}^{2}}\partial_{\hat{v}}\hat{X}_{T}^{(1)}+\frac{f(\hat{z})}{\hat{z}^{2}}\partial_{\hat{z}}\hat{X}\right)\delta\hat{X}_{T}^{(1)}\bigg{|}_{0_{2}}^{\hat{z}_{+}}\right\\},$
(3.2)

where the linearized EOM for $\hat{X}_{T}^{(1)}$ (cf. (2.25)) is used. It is
natural to assume that $\hat{X}_{T}^{(1)}$ is continuous across the cutting
point:

$\displaystyle\hat{X}_{T}^{(1)}(\hat{z}_{+})=\hat{X}_{T}^{(1)}(\hat{z}_{-}).$
(3.3)

Then, the extremum condition at the cutting point gives rise to

$\displaystyle\frac{\delta S_{\rm NG}^{(2),\rm
T}}{\delta\hat{X}_{T}^{(1)}(\hat{z}_{h}+\epsilon,\hat{v})}=0\Longrightarrow
f(\hat{z})\partial_{\hat{z}}\hat{X}_{T}^{(1)}\big{|}_{\hat{z}=\hat{z}_{+}}=f(\hat{z})\partial_{\hat{z}}\hat{X}_{T}^{(1)}\big{|}_{\hat{z}=\hat{z}_{-}}.$
(3.4)

Since $f(\hat{z})$ vanishes at the horizon, the gluing condition implies
$\partial_{\hat{z}}\hat{X}_{T}^{(1)}$ may not be continuous across the
horizon.

The same analysis can be made for the longitudinal mode $\hat{X}_{L}^{(1)}$,
which will give the same gluing conditions as (3.3) and (3.4):

$\displaystyle\hat{X}_{T}^{(1)}(\hat{z}_{+})=\hat{X}_{T}^{(1)}(\hat{z}_{-}),\qquad
f(\hat{z})\partial_{\hat{z}}\hat{X}_{L}^{(1)}\big{|}_{\hat{z}=\hat{z}_{+}}=f(\hat{z})\partial_{\hat{z}}\hat{X}_{L}^{(1)}\big{|}_{\hat{z}=\hat{z}_{-}}.$
(3.5)

Physically, the gluing conditions (3.3), (3.4) and (3.5) correspond to
continuity for string’s momentum. We will use them to glue the piecewise
solutions to be found for the upper segment and the lower segment in next two
subsections.

###  3.2 From linearly independent solutions to solution on the contour

In this subsection, we search for linearly independent solutions for
$\hat{X}_{T,L}^{(1)}$ on upper or lower segments of the contour. Then, the
general solution (superposition of linearly independent solutions) will be
glued at the cutting point, under the conditions (3.3), (3.4) and (3.5).
Eventually, the AdS boundary conditions (2.26) will determine all the
integration constants.

It is more convenient to turn to frequency domain

$\displaystyle\Phi(\hat{z},\hat{v})=\int\frac{d\hat{\omega}}{2\pi}e^{-i\hat{\omega}\hat{v}}\Phi(\hat{z},\hat{\omega}),$
(3.6)

where the Fourier mode $\Phi(\hat{z},\hat{\omega})$ satisfies a homogeneous
ordinary differential equation (ODE):

$\displaystyle
0=\partial_{\hat{z}}\left[\frac{f(\hat{z})}{\hat{z}^{2}}\partial_{\hat{z}}\Phi\right]+\frac{2}{\hat{z}^{2}}i\hat{\omega}\partial_{\hat{z}}\Phi-\frac{2}{\hat{z}^{3}}i\hat{\omega}\Phi.$
(3.7)

Apparently, the time-reversal symmetry is not realized simply as

$\displaystyle\hat{\omega}\to-\hat{\omega},\qquad\Phi(\hat{z},\hat{\omega})\to\Phi(\hat{z},-\hat{\omega}).$
(3.8)

In order to see how the time-reversal symmetry is realized, we temporarily
turn to Schwarzschild coordinate $(\hat{z},\hat{t})$ for the worldsheet space,
in which the linearized string fluctuations
$\hat{X}_{T,L}^{(1)}(\hat{z},\hat{t})$ will be collectively denoted as
$\tilde{\Phi}(\hat{z},\hat{t})$. In the frequency domain:

$\displaystyle\tilde{\Phi}(\hat{z},\hat{t})=\int\frac{d\hat{\omega}}{2\pi}e^{-i\omega\hat{t}}\tilde{\Phi}(\hat{z},\omega),$
(3.9)

where the Fourier mode $\tilde{\Phi}(\hat{z},\hat{\omega})$ obeys

$\displaystyle
0=\partial_{\hat{z}}\left[\frac{f(\hat{z})}{\hat{z}^{2}}\partial_{\hat{z}}\tilde{\Phi}\right]+\frac{\hat{\omega}^{2}}{\hat{z}^{2}f(\hat{z})}\tilde{\Phi}.$
(3.10)

Here, time-reversal symmetry is simply realized as

$\displaystyle\hat{\omega}\to-\hat{\omega},\qquad\tilde{\Phi}(\hat{z},\hat{\omega})\to\tilde{\Phi}(\hat{z},-\hat{\omega}),$
(3.11)

which will be helpful for finding linearly independent solutions.

In the single copy of the contour, namely, $\hat{z}\in[0,z_{h}]$, the EOM
(3.10) admits two linearly independent solutions: the ingoing mode and
outgoing mode. Near the horizon, the ingoing mode behaves as

$\displaystyle\tilde{\Phi}^{\rm ig}(\hat{z}\to
z_{h},\hat{\omega})=(z_{h}-\hat{z})^{-i\hat{\omega}/(4\pi
T)}\left[\tilde{\Phi}_{h}^{0}+\tilde{\Phi}_{h}^{1}(z_{h}-\hat{z})+\tilde{\Phi}_{h}^{2}(z_{h}-\hat{z})^{2}+\cdots\right],$
(3.12)

where $\tilde{\Phi}_{h}^{1},\tilde{\Phi}_{h}^{2},\cdots$ are determined in
terms of $\tilde{\Phi}_{h}^{0}$, and the horizon data $\tilde{\Phi}_{h}^{0}$
is undetermined. The ingoing solution $\tilde{\Phi}^{\rm ig}$ is uniquely
fixed once the horizon data $\tilde{\Phi}_{h}^{0}$ is given. Indeed, the
horizon data $\tilde{\Phi}_{h}^{0}$ is in one-to-one correspondence to the
boundary value $\tilde{\Phi}^{\rm ig}(\hat{z}=0,\hat{\omega})$.

Under the time-reversal symmetry (3.11), the outgoing mode is

$\displaystyle\tilde{\Phi}^{\rm og}(\hat{z},\hat{\omega})=\tilde{\Phi}^{\rm
ig}(\hat{z},-\hat{\omega}).$ (3.13)

Now, we would like to convert the relationship (3.13) into the ingoing EF
coordinate. This is achieved by realizing that $\Phi$ (also $\tilde{\Phi}$) is
a scalar in the worldsheet space, which tells that

$\displaystyle\Phi(\hat{z},\hat{\omega})e^{-i\hat{\omega}\hat{v}}=\tilde{\Phi}(\hat{z},\hat{\omega})e^{-i\hat{\omega}\hat{t}}\Longrightarrow\Phi(\hat{z},\hat{\omega})=\tilde{\Phi}(\hat{z},\hat{\omega})e^{i\hat{\omega}\chi_{s}(\hat{z})},\qquad
s=1,2,$ (3.14)

where the subscript “$s$” is to remind the upper or lower segment. Here, we
made use of the relation (2.16). Notice that the relationship (3.14) also
holds for each linearly independent solution. So, in the ingoing EF
coordinate, an analogue of (3.13) is

$\displaystyle\Phi^{\rm
og}(\hat{z},\hat{\omega})=e^{2i\hat{\omega}\chi_{s}(\hat{z})}\Phi^{\rm
ig}(\hat{z},-\hat{\omega}),$ (3.15)

which could interpreted as nonlinear realization of time-reversal symmetry
[44].

Thus, we immediately write down general solutions on the upper and lower
segments:

$\displaystyle\Phi^{\rm up}(\hat{z},\hat{\omega})$ $\displaystyle=c^{\rm
up}\Phi^{\rm ig}(\hat{z},\hat{\omega})+d^{\rm
up}e^{2i\hat{\omega}\chi_{2}(\hat{z})}\Phi^{\rm
ig}(\hat{z},-\hat{\omega}),\,\,\qquad\hat{z}\in[0_{2},z_{h}+\epsilon],$
$\displaystyle\Phi^{\rm dw}(\hat{z},\hat{\omega})$ $\displaystyle=c^{\rm
dw}\Phi^{\rm ig}(\hat{z},\hat{\omega})+d^{\rm
dw}e^{2i\hat{\omega}\chi_{1}(\hat{z})}\Phi^{\rm
ig}(\hat{z},-\hat{\omega}),\qquad\hat{z}\in[0_{1},z_{h}+\epsilon].$ (3.16)

Here, since the ingoing solution $\Phi^{\rm ig}$ is regular, we do not need to
distinguish between the upper and lower segments. Imposing the gluing
conditions (3.3), (3.4) and (3.5) on the solutions (3.16), we obtain

$\displaystyle c^{\rm up}=c^{\rm dw},\qquad d^{\rm up}=d^{\rm
dw}e^{\beta_{0}\hat{\omega}},$ (3.17)

where $\beta_{0}$ is the inverse temperature $1/T$. So, the general solutions
(3.16) become

$\displaystyle\Phi^{\rm up}(\hat{z},\hat{\omega})$ $\displaystyle=c\Phi^{\rm
ig}(\hat{z},\hat{\omega})+de^{2i\hat{\omega}\chi_{2}(\hat{z})}\Phi^{\rm
ig}(\hat{z},-\hat{\omega}),\qquad\qquad\hat{z}\in[0_{2},z_{h}+\epsilon],$
$\displaystyle\Phi^{\rm dw}(\hat{z},\hat{\omega})$ $\displaystyle=c\Phi^{\rm
ig}(\hat{z},\hat{\omega})+de^{-\beta\hat{\omega}}e^{2i\hat{\omega}\chi_{1}(\hat{z})}\Phi^{\rm
ig}(\hat{z},-\hat{\omega}),\qquad\hat{z}\in[0_{1},z_{h}+\epsilon],$ (3.18)

where a relabelling $c^{\rm up}\to c,d^{\rm up}\to d$ was made. The piecewise
solutions (3.18) could be put into a compact form:

$\displaystyle\Phi(\hat{z},\hat{\omega})$ $\displaystyle=c\Phi^{\rm
ig}(\hat{z},\hat{\omega})+de^{2i\hat{\omega}\chi(\hat{z})}\Phi^{\rm
ig}(\hat{z},-\hat{\omega}),\qquad\qquad\hat{z}\in[0_{1},0_{2}],$ (3.19)

where the function $\chi(\hat{z})$ is defined over the entire contour:

$\displaystyle\chi(\hat{z})\equiv-\int_{0_{2}}^{\hat{z}}\frac{dy}{f(y)},\qquad\hat{z}\in[0_{1},0_{2}].$
(3.20)

The integration constants $c,d$ are fixed by the AdS boundary conditions:

$\displaystyle c\Phi^{\rm ig(0)}(\hat{\omega})+d\Phi^{\rm
ig(0)}(-\hat{\omega})=\hat{q}_{2}(\hat{\omega}),\qquad c\Phi^{\rm
ig(0)}(\hat{\omega})+de^{-\beta_{0}\hat{\omega}}\Phi^{\rm
ig(0)}(-\hat{\omega})=\hat{q}_{1}(\hat{\omega})$ $\displaystyle\Rightarrow$
$\displaystyle
c=\frac{1}{2}\coth\frac{\beta_{0}\hat{\omega}}{2}\frac{\hat{q}_{a}(\hat{\omega})}{\Phi^{\rm
ig(0)}(\hat{\omega})}+\frac{\hat{q}_{r}(\hat{\omega})}{\Phi^{\rm
ig(0)}(\hat{\omega})},\qquad
d=-\frac{\hat{q}_{a}(\hat{\omega})}{(1-e^{-\beta_{0}\hat{\omega}})\Phi^{\rm
ig(0)}(-\hat{\omega})},$ (3.21)

where $\Phi^{\rm ig(0)}(\hat{\omega})=\Phi^{\rm ig}(\hat{z}=0,\hat{\omega})$.
Here, we introduced the $(r,a)$-basis:

$\displaystyle\hat{q}_{r}=\frac{1}{2}(\hat{q}_{1}+\hat{q}_{2}),\qquad\qquad\hat{q}_{a}=\hat{q}_{1}-\hat{q}_{2}.$
(3.22)

Recovering the superscripts “$T,L$” in the position $\hat{q}_{r},\hat{q}_{a}$,
we obtain explicit solutions for linearized string perturbation
$\hat{X}_{T,L}^{(1)}$ over the entire contour:

$\displaystyle\hat{X}_{T,L}^{(1)}(\hat{z},\hat{\omega})=A(\hat{z},\hat{\omega})\hat{q}_{r}^{T,L}(\hat{\omega})+B(\hat{z},\hat{\omega})\hat{q}_{a}^{T,L}(\hat{\omega}),\qquad\hat{z}\in[0_{1},0_{2}],$
(3.23)

where

$\displaystyle A(\hat{z},\hat{\omega})=\frac{\Phi^{\rm
ig}(\hat{z},\hat{\omega})}{\Phi^{\rm ig(0)}(\hat{\omega})},\qquad
B(\hat{z},\hat{\omega})=\frac{1}{2}\coth\frac{\beta_{0}\hat{\omega}}{2}\frac{\Phi^{\rm
ig}(\hat{z},\hat{\omega})}{\Phi^{\rm
ig(0)}(\hat{\omega})}-\frac{e^{2i\hat{\omega}\chi(\hat{z})}}{1-e^{-\beta_{0}\hat{\omega}}}\frac{\Phi^{\rm
ig}(\hat{z},-\hat{\omega})}{\Phi^{\rm ig(0)}(-\hat{\omega})}.$ (3.24)

Thus, solving linearized string perturbation over the entire contour boils
down to searching for ingoing solution $\Phi^{\rm ig}(\hat{z},\hat{\omega})$
in single copy of the radial contour.

##  4 Schwinger-Keldysh effective action for Brownian particle

In this section, we plug the solution for linearized string perturbation of
section 3 into $S_{\rm NG}^{(2),\rm P1}$ (2.31) and $S_{\rm NG}^{(3)}$ (2.22),
and evaluate the radial integral, producing the SK effective action for heavy
quark:

$\displaystyle S_{eff}^{(2)}=S_{\rm NG}^{(2),\rm P1},\qquad\qquad\qquad
S_{eff}^{(3)}=S_{\rm NG}^{(3)}\big{|}_{\hat{X}_{T,L}\to\hat{X}^{(1)}_{T,L}}.$
(4.1)

###  4.1 Quadratic action: coloured noise

In order to compute the quadratic order action (2.31), we need near-boundary
expansion for $e^{2i\hat{\omega}\chi_{1,2}(\hat{z})}$:

$\displaystyle e^{2i\hat{\omega}\chi_{s}(\hat{z})}\xrightarrow[]{\hat{z}\to
0_{s}}1-2i\hat{\omega}\hat{z}-2\hat{\omega}^{2}\hat{z}^{2}+\frac{4}{3}i\hat{\omega}^{3}\hat{z}^{3}+\cdots,\qquad
s=1~{}~{}{\rm or}~{}~{}2.$ (4.2)

Near the AdS boundary, the ingoing mode $\Phi^{\rm ig}$ behaves as

$\displaystyle\Phi^{\rm ig}(\hat{z}\to 0,\hat{\omega})=\Phi^{\rm
ig(0)}(\hat{\omega})-i\hat{\omega}\Phi^{\rm
ig(0)}(\hat{\omega})\hat{z}+\Phi^{\rm ig(3)}(\hat{\omega})\hat{z}^{3}+\cdots.$
(4.3)

Then, the near-boundary behaviors of $\hat{X}_{T,L}^{(1)}$ are

$\displaystyle\hat{X}_{T,L}^{(1)}(\hat{z}\to
0_{s},\hat{\omega})=\hat{q}_{s}^{T,L}(\hat{\omega})-i\hat{\omega}\hat{q}_{s}^{T,L}(\hat{\omega})\hat{z}+\mathbb{O}^{T,L}_{s}(\hat{\omega})\hat{z}^{3}+\cdots,\qquad
s=1~{}~{}{\rm or}~{}~{}2,$ (4.4)

where the normalizable modes $\mathbb{O}^{T,L}_{s}(\hat{\omega})$ are
expressed in terms of ratio $\Phi^{\rm ig(3)}/\Phi^{\rm ig(0)}$:

$\displaystyle\mathbb{O}^{T,L}_{2}(\hat{\omega})=$
$\displaystyle\frac{\Phi^{\rm ig(3)}(\hat{\omega})}{\Phi^{\rm
ig(0)}(\hat{\omega})}\hat{q}_{r}^{T,L}(\hat{\omega})+\frac{1}{2}\coth\frac{\beta_{0}\hat{\omega}}{2}\frac{\Phi^{\rm
ig(3)}(\hat{\omega})}{\Phi^{\rm
ig(0)}(\hat{\omega})}\hat{q}_{a}^{T,L}(\hat{\omega})$
$\displaystyle-\frac{1}{1-e^{-\beta_{0}\hat{\omega}}}\frac{\Phi^{\rm
ig(3)}(-\hat{\omega})}{\Phi^{\rm
ig(0)}(-\hat{\omega})}\hat{q}_{a}^{T,L}(\hat{\omega})+\frac{2i\hat{\omega}^{3}}{3(1-e^{-\beta_{0}\hat{\omega}})}\hat{q}_{a}^{T,L}(\hat{\omega}),$
$\displaystyle\mathbb{O}^{T,L}_{1}(\hat{\omega})=$
$\displaystyle\frac{\Phi^{\rm ig(3)}(\hat{\omega})}{\Phi^{\rm
ig(0)}(\hat{\omega})}\hat{q}_{r}^{T,L}(\hat{\omega})+\frac{1}{2}\coth\frac{\beta_{0}\hat{\omega}}{2}\frac{\Phi^{\rm
ig(3)}(\hat{\omega})}{\Phi^{\rm
ig(0)}(\hat{\omega})}\hat{q}_{a}^{T,L}(\hat{\omega})$
$\displaystyle-\frac{e^{-\beta_{0}\hat{\omega}}}{1-e^{-\beta_{0}\hat{\omega}}}\frac{\Phi^{\rm
ig(3)}(-\hat{\omega})}{\Phi^{\rm
ig(0)}(-\hat{\omega})}\hat{q}_{a}^{T,L}(\hat{\omega})+\frac{2i\hat{\omega}^{3}e^{-\beta_{0}\hat{\omega}}}{3(1-e^{-\beta_{0}\hat{\omega}})}\hat{q}_{a}^{T,L}(\hat{\omega}).$
(4.5)

Plugging (4.4) into (2.31), we obtain the quadratic order SK effective action

$\displaystyle S_{eff}^{(2)}$
$\displaystyle=\frac{1}{2\pi\alpha^{\prime}}\int\frac{d\hat{\omega}}{2\pi}\left\\{\frac{i}{2}\hat{q}_{a}^{T}(-\hat{\omega})\hat{G}_{rr}(\hat{\omega})\hat{q}_{a}^{T}(\hat{\omega})+\hat{q}_{a}^{T}(-\hat{\omega})\left[\hat{M}\hat{\omega}^{2}+\hat{G}_{ra}(\hat{\omega})\right]\hat{q}_{r}^{T}(\hat{\omega})\right\\}$
$\displaystyle+\frac{\gamma^{2}}{2\pi\alpha^{\prime}}\int\frac{d\hat{\omega}}{2\pi}\left\\{\frac{i}{2}\hat{q}_{a}^{L}(-\hat{\omega})\hat{G}_{rr}(\hat{\omega})\hat{q}_{a}^{L}(\hat{\omega})+\hat{q}_{a}^{L}(-\hat{\omega})\left[\hat{M}\hat{\omega}^{2}+\hat{G}_{ra}(\hat{\omega})\right]\hat{q}_{r}^{L}(\hat{\omega})\right\\},$
(4.6)

where

$\displaystyle\hat{G}_{rr}(\hat{\omega})=-i\coth\frac{\beta_{0}\hat{\omega}}{2}\left[\frac{3}{2}\frac{\Phi^{\rm
ig(3)}(\hat{\omega})}{\Phi^{\rm
ig(0)}(\hat{\omega})}-\frac{3}{2}\frac{\Phi^{\rm
ig(3)}(-\hat{\omega})}{\Phi^{\rm
ig(0)}(-\hat{\omega})}+i\hat{\omega}^{3}\right],$
$\displaystyle\hat{G}_{ra}(\hat{\omega})=3\frac{\Phi^{\rm
ig(3)}(\hat{\omega})}{\Phi^{\rm ig(0)}(\hat{\omega})}+i\hat{\omega}^{3}.$
(4.7)

The quadratic action (4.6) was previously obtained in [41, 42] by using
Kruskal extension of the AdS/CFT correspondence [39]. In terms of the rescaled
variables $\hat{q}^{T,L}$, (4.6) looks essentially the same as that of static
case [5]. In the Schwarzschild coordinate, the $i\hat{\omega}^{3}$ piece in
$\hat{G}_{rr}$ and $\hat{G}_{ra}$ will be absorbed into the normalizable mode
$\tilde{\Phi}^{\rm ig(3)}$:

$\displaystyle\frac{\Phi^{\rm ig(3)}(\hat{\omega})}{\Phi^{\rm
ig(0)}(\hat{\omega})}=\frac{\tilde{\Phi}^{\rm
ig(3)}(\hat{\omega})}{\tilde{\Phi}^{\rm
ig(0)}(\hat{\omega})}-\frac{1}{3}i\hat{\omega},$ (4.8)

which will render the correlators (4.7) to be of the same form as in [5] (see
(A11) of [5]). In (4.6), the $\hat{M}\hat{\omega}^{2}$-term arises from the
divergent part of the AdS boundary limit $\hat{z}\to 0$ [5]:
$\hat{M}=\lim_{\hat{z}\to 0}1/{\hat{z}}$, which is related to zero-temperature
quark mass $M=\lim_{z\to 0}1/z$ [30] by the boost factor:
$\hat{M}=M/\sqrt{\gamma}$. Apparently, the quark mass becomes finite if the
quark lives on a finite cutoff slice $z=\Lambda\neq 0$.

In SK effective action, the quark position $q$ couples to plasma medium force
$\mathcal{F}$ in the form
$q_{1}\mathcal{F}_{1}-q_{2}\mathcal{F}_{2}=q_{r}\mathcal{F}_{a}+q_{a}\mathcal{F}_{r}$
[5]. Thus, $S_{eff}$ acts as generating functional $W$ for correlators of
medium force $\mathcal{F}$:

$\displaystyle
W[\hat{q}_{r}^{L,T},\hat{q}_{a}^{L,T}]=iS_{eff}[\hat{q}_{r}^{L,T},\hat{q}_{a}^{L,T}],$
(4.9)

which, for convenience, is presented in terms of rescaled quantities. Then, we
could interpret $\hat{G}_{rr}$ and $\hat{G}_{ra}$ as the symmetric and
retarded force-force333Here, the force should be understood as rescaled one
$\hat{\mathcal{F}}=\mathcal{F}/\sqrt{\gamma}.$ correlators [5]. Obviously, the
following relation holds

$\displaystyle\hat{G}_{rr}(\hat{\omega})=\coth\frac{\beta_{0}\hat{\omega}}{2}{\rm
Im}[\hat{G}_{ra}(\hat{\omega})],$ (4.10)

which is more easily understood from worldsheet perspective. The relation
(4.10) arises from the following fact: the state described by (2.15) is
thermal, and the theory (2.21) preserves $\mathcal{T}$-symmetry. In terms of
physical quantities by $\hat{\omega}\to\omega,\hat{q}\to
q,\hat{\mathcal{F}}\to\mathcal{F}$, (4.10) will not take the exact form of KMS
condition for a thermal state, which implies the fluctuation and dissipation
felt by relativistic heavy quark is of non-thermal nature [41].

Via relabelling $\hat{\omega}\to-\hat{\omega}$ in the $ra$-term of (4.6), we
can read off advanced correlator

$\displaystyle\hat{G}_{ar}(\hat{\omega})=3\frac{\Phi^{\rm
ig(3)}(-\hat{\omega})}{\Phi^{\rm ig(0)}(-\hat{\omega})}-i\hat{\omega}^{3},$
(4.11)

which obeys the Onsager relation
$[\hat{G}_{ra}(\hat{\omega})]^{*}=\hat{G}_{ar}(\hat{\omega})$.

From (4.7), via Legendre transformation (see [5, 17] for details), one can
recover stochastic equation for heavy quark (here we set
$2\pi\alpha^{\prime}=1$ for convenience):

$\displaystyle\left[\hat{M}\hat{\omega}^{2}+\hat{G}_{ra}(\hat{\omega})\right]\hat{q}_{r}^{T}(\hat{\omega})=\hat{\xi}^{T}(\hat{\omega}),\qquad\qquad\langle\hat{\xi}^{T}(-\hat{\omega})\hat{\xi}^{T}(\hat{\omega})\rangle=\hat{G}_{rr}(\hat{\omega}),$
$\displaystyle\left[\hat{M}\hat{\omega}^{2}+\hat{G}_{ra}(\hat{\omega})\right]\hat{q}_{r}^{L}(\hat{\omega})=\hat{\xi}^{L}(\hat{\omega}),\qquad\qquad\langle\hat{\xi}^{L}(-\hat{\omega})\hat{\xi}^{L}(\hat{\omega})\rangle=\gamma^{-2}\hat{G}_{rr}(\hat{\omega}),$
(4.12)

where
$\hat{\xi}^{T}(\hat{\omega})=-i\hat{G}_{rr}(\hat{\omega})\hat{q}_{a}^{T}(\hat{\omega})$,
and
$\hat{\xi}^{L}(\hat{\omega})=-i\hat{G}_{rr}(\hat{\omega})\hat{q}_{a}^{L}(\hat{\omega})$.
Here, (4.12) generalizes the Langevin equation (1.1), particularly the noise
$\hat{\xi}^{T,L}$ becomes coloured. It is straightforward to rewrite these
equations in terms of original variables (say, $\omega,q^{T,L},\xi^{T,L}$) or
in time domain, see [41, 42] for more details.

Below, we present our results for the retarded correlator
$\hat{G}_{ra}(\hat{\omega})$ either when $\hat{\omega}$ is small
(analytically) or when $\hat{\omega}$ is arbitrary (numerically).

$\bullet$ Low frequency limit

First, we consider the low frequency limit $\hat{\omega}/T\ll 1$ so that

$\displaystyle\Phi^{\rm ig}(\hat{z},\hat{\omega})=\Phi^{\rm
ig}_{0}(\hat{z},\hat{\omega})+\lambda^{1}\Phi^{\rm
ig}_{1}(\hat{z},\hat{\omega})+\lambda^{2}\Phi^{\rm
ig}_{2}(\hat{z},\hat{\omega})+\cdots,$ (4.13)

where the bookkeeping parameter $\lambda\sim\hat{\omega}/T$. Besides
regularity requirement at the horizon for $\Phi^{\rm ig}_{n}$
$(n=0,1,2,\cdots)$, we impose AdS boundary condition:

$\displaystyle\Phi^{\rm
ig}_{0}(\hat{z}=0,\hat{\omega})=1,\qquad\qquad\Phi^{\rm
ig}_{n}(\hat{z}=0,\hat{\omega})=0\qquad{\rm for}~{}~{}~{}n\geq 1.$ (4.14)

In the low frequency limit $\hat{\omega}/T<<1$, we are able to obtain the
ingoing mode $\Phi^{\rm ig}$ up to third order
$\mathcal{O}(\hat{\omega}^{3})$:

$\displaystyle\Phi^{\rm ig}_{0}(\hat{z},\hat{\omega})$ $\displaystyle=1,$
$\displaystyle\Phi^{\rm ig}_{1}(\hat{z},\hat{\omega})$
$\displaystyle=\int_{0}^{\hat{z}}\frac{x^{2}dx}{f(x)}\int_{z_{h}}^{x}\frac{2i\hat{\omega}}{y^{3}}dy=-\frac{i\hat{\omega}}{\pi
T}\arctan(\pi Tz)$ $\displaystyle\xrightarrow[]{\hat{z}\to
0}-i\hat{\omega}\hat{z}+\frac{1}{3}i\hat{\omega}\pi^{2}T^{2}\hat{z}^{3}+\cdots,$
$\displaystyle\Phi^{\rm ig}_{2}(\hat{z},\hat{\omega})$
$\displaystyle=\int_{0}^{\hat{z}}\frac{x^{2}dx}{f(x)}\int_{z_{h}}^{x}\left[-\frac{2}{y^{2}}i\hat{\omega}\partial_{y}\Phi^{\rm
ig}_{1}(y,\hat{\omega})+\frac{2}{y^{3}}i\hat{\omega}\Phi^{\rm
ig}_{1}(y,\hat{\omega})\right]dy$ $\displaystyle=-\frac{\hat{\omega}^{2}}{8\pi
T^{2}}\left[-4\arctan(\pi T\hat{z})+4\left(\arctan(\pi
T\hat{z})\right)^{2}+2\log\frac{(1+\pi
T\hat{z}^{2})^{2}}{1+\pi^{2}T^{2}\hat{z}^{2}}\right]$
$\displaystyle\xrightarrow[]{\hat{z}\to 0}-\frac{1}{3}\hat{\omega}^{2}\pi
T\hat{z}^{3}+\cdots,$ $\displaystyle\Phi^{\rm ig}_{3}(\hat{z},\hat{\omega})$
$\displaystyle=\int_{0}^{\hat{z}}\frac{x^{2}dx}{f(x)}\int_{z_{h}}^{x}\left[-\frac{2}{y^{2}}i\hat{\omega}\partial_{y}\Phi^{\rm
ig}_{2}(y,\hat{\omega})+\frac{2}{y^{3}}i\hat{\omega}\Phi^{\rm
ig}_{2}(y,\hat{\omega})\right]dy$ $\displaystyle\xrightarrow[]{\hat{z}\to
0}\frac{1}{12}i\hat{\omega}^{3}(-4+\pi-2\log 2)\hat{z}^{3}+\cdots.$ (4.15)

Accordingly, the retarded two-point correlator $\hat{G}_{ra}$ is expanded as

$\displaystyle\hat{G}_{ra}(\hat{\omega})=i\hat{\omega}\pi^{2}T^{2}-\hat{\omega}^{2}\pi
T+\frac{1}{4}i\hat{\omega}^{3}(\pi-2\log 2)+\cdots,$ (4.16)

where the first two terms were reported before in [5], and the
$\hat{\omega}^{3}$-term is consistent with relevant numerical result of [44].

$\bullet$ Beyond low frequency limit

For generic frequency $\hat{\omega}$, we numerically solve the EOM (3.7) under
regularity condition. In the ingoing EF coordinate, the analogue of (3.12) is

$\displaystyle\Phi^{\rm ig}(\hat{z}\to
z_{h},\hat{\omega})=\Phi_{h}^{0}+\Phi_{h}^{1}(z_{h}-\hat{z})+\Phi_{h}^{2}(z_{h}-\hat{z})^{2}+\Phi_{h}^{3}(z_{h}-\hat{z})^{3}+\cdots,$
(4.17)

where $\Phi_{h}^{1},\Phi_{h}^{2},\Phi_{h}^{3},\cdots$ are fixed in terms of
the horizon data $\Phi_{h}^{0}$. In practical calculation, we will take
$\Phi_{h}^{0}=1$ so that the near horizon expansion (4.17) provides sufficient
initial conditions for the EOM (3.7). Our numerical results for
$\hat{G}_{ra}(\hat{\omega})$ and $\hat{G}_{rr}(\hat{\omega})$ are shown in
Figure 3, which are in perfect agreement with [59]. While ${\rm
Re}[\hat{G}_{ra}]$ demonstrates a damped oscillating behavior as
$\hat{\omega}$ gets increased, ${\rm Im}[\hat{G}_{ra}]$ (also $\hat{G}_{rr}$)
shows a cubic growing behavior at large values of $\hat{\omega}$.

Figure 3: Retarded and symmetric two-point correlators. Here, we set $\pi T=1$
for convenience.

###  4.2 Cubic action: non-Gaussianity

In this subsection, we compute cubic order effective action via replacing
$\hat{X}_{T,L}$ in (2.22) by the linearized string perturbation (3.23) and
then evaluating the radial integral. In the frequency domain, the cubic order
action becomes

$\displaystyle S_{\rm
NG}^{(3)}=-\frac{\beta\gamma}{2\pi\alpha^{\prime}}\int\frac{d\hat{\omega}_{2}d\hat{\omega}_{3}}{(2\pi)^{2}}\delta(\hat{\omega}_{1}+\hat{\omega}_{2}+\hat{\omega}_{3})\sum_{i=1}^{7}I_{i}(\hat{\omega}_{1},\hat{\omega}_{2},\hat{\omega}_{3}),$
(4.18)

where

$\displaystyle I_{1}$
$\displaystyle=\int_{0_{1}}^{0_{2}}d\hat{z}\frac{\gamma^{2}\pi^{2}T^{2}}{2}f(\hat{z})\partial_{\hat{z}}\hat{X}_{L}^{(1)}(\hat{z},\hat{\omega}_{1})\partial_{\hat{z}}\hat{X}_{L}^{(1)}(\hat{z},\hat{\omega}_{2})\partial_{\hat{z}}\hat{X}_{L}^{(1)}(\hat{z},\hat{\omega}_{3}),$
$\displaystyle I_{2}$
$\displaystyle=\int_{0_{1}}^{0_{2}}d\hat{z}\frac{\pi^{2}T^{2}}{2}f(\hat{z})\partial_{\hat{z}}\hat{X}_{L}^{(1)}(\hat{z},\hat{\omega}_{1})\partial_{\hat{z}}\hat{X}_{T}^{(1)}(\hat{z},\hat{\omega}_{2})\partial_{\hat{z}}\hat{X}_{T}^{(1)}(\hat{z},\hat{\omega}_{3}),$
$\displaystyle I_{3}$
$\displaystyle=\int_{0_{1}}^{0_{2}}d\hat{z}\frac{-i\hat{\omega}_{3}\gamma^{2}}{2\hat{z}^{2}}(1-3\pi^{2}T^{2}\hat{z}^{2})\partial_{\hat{z}}\hat{X}_{L}^{(1)}(\hat{z},\hat{\omega}_{1})\partial_{\hat{z}}\hat{X}_{L}^{(1)}(\hat{z},\hat{\omega}_{2})\hat{X}_{L}^{(1)}(\hat{z},\hat{\omega}_{3}),$
$\displaystyle I_{4}$
$\displaystyle=\int_{0_{1}}^{0_{2}}d\hat{z}\frac{i\hat{\omega}_{3}}{2\hat{z}^{2}}(1+\pi^{2}T^{2}\hat{z}^{2})\partial_{\hat{z}}\hat{X}_{T}^{(1)}(\hat{z},\hat{\omega}_{1})\partial_{\hat{z}}\hat{X}_{T}^{(1)}(\hat{z},\hat{\omega}_{2})\hat{X}_{L}^{(1)}(\hat{z},\hat{\omega}_{3}),$
$\displaystyle I_{5}$
$\displaystyle=\int_{0_{1}}^{0_{2}}d\hat{z}\frac{-i\hat{\omega}_{3}f(\hat{z})}{\hat{z}^{2}(1+\pi^{2}T^{2}\hat{z}^{2})}\partial_{\hat{z}}\hat{X}_{T}^{(1)}(\hat{z},\hat{\omega}_{1})\partial_{\hat{z}}\hat{X}_{L}^{(1)}(\hat{z},\hat{\omega}_{2})\hat{X}_{T}^{(1)}(\hat{z},\hat{\omega}_{3}),$
$\displaystyle I_{6}$
$\displaystyle=\int_{0_{1}}^{0_{2}}d\hat{z}\frac{\hat{\omega}_{2}\hat{\omega}_{3}\gamma^{2}}{\hat{z}^{2}(1+\pi^{2}T^{2}\hat{z}^{2})}\partial_{\hat{z}}\hat{X}_{L}^{(1)}(\hat{z},\hat{\omega}_{1})\hat{X}_{L}^{(1)}(\hat{z},\hat{\omega}_{2})\hat{X}_{L}^{(1)}(\hat{z},\hat{\omega}_{3}),$
$\displaystyle I_{7}$
$\displaystyle=\int_{0_{1}}^{0_{2}}d\hat{z}\frac{\hat{\omega}_{2}\hat{\omega}_{3}}{\hat{z}^{2}(1+\pi^{2}T^{2}\hat{z}^{2})}\partial_{\hat{z}}\hat{X}_{L}^{(1)}(\hat{z},\hat{\omega}_{1})\hat{X}_{T}^{(1)}(\hat{z},\hat{\omega}_{2})\hat{X}_{T}^{(1)}(\hat{z},\hat{\omega}_{3}).$
(4.19)

Our primary interest is to derive $S_{eff}^{(3)}$ up to first order in time
derivative, which amounts to computing (4.19) to the order
$\mathcal{O}(\hat{\omega}^{1})$. However, in order to discuss nonlinear KMS
condition, we will also track part of $\mathcal{O}(\hat{\omega}^{2})$-terms in
$S_{eff}^{(3)}$, say $arr$-type terms. For a $\mathcal{PT}$-invariant theory
in a thermal state, the KMS condition for generating functional sets four
constraints among the full set of three-point correlators [57]: they involve
all $\mathcal{O}(\hat{\omega}^{1})$-terms in the cubic action, and additional
$arr$-terms at the order $\mathcal{O}(\hat{\omega}^{2})$, see appendix C for a
summary on KMS constraints for three-point functions.

There is a subtle issue arising from the $\hat{\omega}$-expansion for
$\hat{X}_{T,L}^{(1)}$ (3.23). While the function $A$ is regular over the
contour, the function $B$ contains singularity (branch cut) near the horizon,
due to the factor $e^{2i\hat{\omega}\chi(\hat{z})}$. It is important to notice
that near the horizon, the limit $\hat{\omega}\to 0$ and the limit
$\epsilon\to 0$ of this factor $e^{2i\hat{\omega}\chi(\hat{z})}$ do not
commute:

$\displaystyle\lim_{\hat{\omega}\to 0}\lim_{\epsilon\to
0}e^{2i\hat{\omega}\chi(\hat{z})}\neq\lim_{\epsilon\to 0}\lim_{\hat{\omega}\to
0}e^{2i\hat{\omega}\chi(\hat{z})},\qquad{\rm as}~{}~{}\hat{z}\to
z_{h}+\epsilon.$ (4.20)

If we expand the factor $e^{2i\hat{\omega}\chi(\hat{z})}$ first in the limit
$\hat{\omega}\to 0$, the nature of horizon singularity for this factor will be
changed. Particularly, in order to guarantee $\hat{\omega}$-expansion to be
convergent, the radius $\epsilon$ of the infinitesimal circle in Figure 2
should not be exponentially small [5, 42, 45]:

$\displaystyle 1\gg\frac{\epsilon}{z_{h}}\gg e^{-l_{\rm mfp}T},$ (4.21)

where $l_{\rm mfp}$ is the typical time scale associated with the variations
of the position $\hat{q}^{T,L}$. However, even when $l_{\rm mfp}T\to+\infty$
the condition (4.21) seems to be in tension with the limit $\epsilon\to 0$.
This subtlety has been avoided in the computation of quadratic order action
$S_{\rm NG}^{(2),\rm P1}$, which is reduced into a boundary term without any
expansion in small $\hat{\omega}$.

We have shown that the subtly (4.20) is accidentally washed away in present
work. In other words, in order to extract
$\mathcal{O}(\hat{\omega}^{1})$-terms and $\mathcal{O}(\hat{\omega}^{2})$
$arr$-terms of (4.19), it becomes legal to expand the integrands of (4.19) in
small $\hat{\omega}$ before $\epsilon\to 0$ is taken. The demonstration is
technical and is deferred to appendix B. We proceed by expanding integrands of
(4.19) in small $\hat{\omega}$ regime.

Up to $\mathcal{O}(\hat{\omega}^{1})$, the functions $A,B$ (3.24) and their
radial derivatives are expanded as:

$\displaystyle A(z,\hat{\omega})=$ $\displaystyle 1-\frac{i\hat{\omega}}{\pi
T}\arctan(\pi T\hat{z})+\cdots,$ $\displaystyle B(\hat{z},\hat{\omega})=$
$\displaystyle\left[-\frac{1}{2}-\frac{2i}{\pi}\arctan(\pi
T\hat{z})-2iT\chi(\hat{z})\right]+\hat{\omega}\left[-\frac{i}{2\pi
T}\arctan(\pi T\hat{z})-i\chi(\hat{z})\right.$
$\displaystyle\left.+\frac{2}{\pi}\arctan(\pi
T\hat{z})\chi(\hat{z})+2T(\chi(\hat{z}))^{2}\right]+\cdots,$
$\displaystyle\partial_{\hat{z}}A(\hat{z},\hat{\omega})=$
$\displaystyle-\frac{i\hat{\omega}}{1+\pi^{2}T^{2}\hat{z}^{2}}+\cdots,$
$\displaystyle\partial_{\hat{z}}B(\hat{z},\hat{\omega})=$
$\displaystyle\frac{2i\pi^{2}T^{3}\hat{z}^{2}}{1-\pi^{4}T^{4}\hat{z}^{4}}+\hat{\omega}\left[\frac{i(1+\pi^{2}T^{2}\hat{z}^{2})}{2(1-\pi^{4}T^{4}\hat{z}^{4})}-\frac{2\arctan(\pi
T\hat{z})}{\pi(1-\pi^{4}T^{4}\hat{z}^{4})}-\frac{2T(1+\pi^{2}T^{2}\hat{z}^{2})}{1-\pi^{4}T^{4}\hat{z}^{4}}\chi(\hat{z})\right]+\cdots,$
(4.22)

where $\chi(\hat{z})$ defined in (3.20) has explicit form:

$\displaystyle\chi(\hat{z})=-\frac{\arctan(\pi T\hat{z})}{2\pi
T}-\frac{\log(1+\pi T\hat{z})}{4\pi T}+\frac{\log(1-\pi T\hat{z})}{4\pi
T},\qquad\hat{z}\in[0_{1},0_{2}].$ (4.23)

Apparently, $\chi(\hat{z})$ has a branch cut near the horizon and thus is a
multi-valued function. In piecewise form, the function $\chi(\hat{z})$ becomes

$\displaystyle\chi(\hat{z})$ $\displaystyle=-\frac{\arctan(\pi T\hat{z})}{2\pi
T}-\frac{\log(1+\pi T\hat{z})}{4\pi T}+\frac{\log(1-\pi T\hat{z})}{4\pi
T}-\frac{i}{2T},\,\qquad\quad\hat{z}\in[0_{1},z_{h}-\epsilon],$
$\displaystyle\chi(\hat{z})$ $\displaystyle=-\frac{\arctan(\pi T\hat{z})}{2\pi
T}-\frac{\log(1+\pi T\hat{z})}{4\pi T}+\frac{\log(1-\pi T\hat{z})}{4\pi
T}\bigg{|}_{\hat{z}=z_{h}+\epsilon e^{i\theta}},\qquad\theta\in[-\pi,\pi],$
$\displaystyle\chi(\hat{z})$ $\displaystyle=-\frac{\arctan(\pi T\hat{z})}{2\pi
T}-\frac{\log(1+\pi T\hat{z})}{4\pi T}+\frac{\log(1-\pi T\hat{z})}{4\pi
T},\qquad\qquad\qquad\hat{z}\in[0_{2},z_{h}-\epsilon],$ (4.24)

Since the integrands of $I_{1},I_{2}$ only involve simple pole at
$\hat{z}=z_{h}$ enclosed by the contour, they could be computed via residue
theorem. The results are

$\displaystyle
I_{1}=\frac{3\gamma^{2}\pi^{2}T^{3}}{2}\left[T\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{L}(\hat{\omega}_{3})+\hat{\omega}_{1}\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{L}(\hat{\omega}_{3})\hat{q}_{r}^{L}(\hat{\omega}_{1})\right],$
$\displaystyle
I_{2}=\frac{3\pi^{2}T^{3}}{2}\left[T\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{T}(\hat{\omega}_{2})\hat{q}_{a}^{T}(\hat{\omega}_{3})+\hat{\omega}_{1}\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{T}(\hat{\omega}_{3})\hat{q}_{r}^{T}(\hat{\omega}_{1})\right].$
(4.25)

The integrals $I_{3,4,5}$ will involve logarithmic branch cut at
$\hat{z}=z_{h}$, and will be computed by splitting the radial contour of
Figure 2 as

$\displaystyle\int_{0_{1}}^{0_{2}}d\hat{z}=\int_{0_{1}}^{z_{h}-\epsilon}d\hat{z}+\int_{\mathcal{C}}d\theta+\int_{z_{h}-\epsilon}^{0_{2}}d\hat{z},$
(4.26)

where the integrals over the two horizontal legs will be partially cancelled.
The integrals $I_{6,7}$ do not contribute at the order
$\mathcal{O}(\hat{\omega}^{1})$. For illustration we take the computation of
$I_{3}$ as an example. For convenience, we split $I_{3}$ into three parts:

$\displaystyle
I_{3}=I_{3}^{(1)}\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{r}^{L}(\hat{\omega}_{3})+I_{3}^{(2)}\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{L}(\hat{\omega}_{3})+I_{3}^{(3)}\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{L}(\hat{\omega}_{3}),$
(4.27)

where $I_{3}^{(1)}$ and $I_{3}^{(2)}$ are computed directly via the residue
theorem:

$\displaystyle I_{3}^{(1)}$
$\displaystyle=\int_{0_{1}}^{0_{2}}d\hat{z}\frac{2i\hat{\omega}_{3}\gamma^{2}\pi^{4}T^{6}\hat{z}^{2}(1-3\pi^{2}T^{2}\hat{z}^{2})}{(1-\pi^{4}T^{4}\hat{z}^{4})^{2}}=\gamma^{2}\pi^{2}T^{3}\hat{\omega}_{3},$
$\displaystyle I_{3}^{(2)}$
$\displaystyle=\int_{0_{1}}^{0_{2}}d\hat{z}\frac{\gamma^{2}\pi^{3}T^{6}\hat{\omega}_{3}\hat{z}^{2}(-1+3\pi^{2}T^{2}\hat{z}^{2})}{(1-\pi^{4}T^{4}\hat{z}^{4})^{2}}\left[i\pi-2\arctan(\pi
T\hat{z})+\log(1+\pi T\hat{z})\right]$
$\displaystyle=-\frac{1}{8}\gamma^{2}\pi
T^{3}\hat{\omega}_{3}\left[i+(4+2i)\pi-4i\log 2\right].$ (4.28)

The third part is

$\displaystyle
I_{3}^{(3)}=\int_{0_{1}}^{0_{2}}d\hat{z}\frac{\gamma^{2}\pi^{3}T^{6}\hat{\omega}_{3}\hat{z}^{2}(1-3\pi^{2}T^{2}\hat{z}^{2})}{(1-\pi^{4}T^{4}\hat{z}^{4})^{2}}\log(1-\pi
T\hat{z}),$ (4.29)

where the integrand involves a branch cut at $\hat{z}=z_{h}$. Thus,
$I_{3}^{(3)}$ will be computed by splitting the contour as in (4.26). The
phase of the argument $\hat{z}$ is chosen as in (4.24). The contribution from
the horizontal legs is

$\displaystyle\int_{0}^{1/(\pi
T)-\epsilon}d\hat{z}\frac{\gamma^{2}\pi^{3}T^{6}\hat{\omega}_{3}\hat{z}^{2}(1-3\pi^{2}T^{2}\hat{z}^{2})}{(1-\pi^{4}T^{4}\hat{z}^{4})^{2}}(-2i\pi)$
$\displaystyle=\frac{i\hat{\omega}_{3}\gamma^{2}T^{2}}{4\epsilon}+\frac{1}{2}i\hat{\omega}_{3}\gamma^{2}\pi
T^{3}\log\epsilon-\frac{1}{8}i\hat{\omega}_{3}\gamma^{2}\pi T^{3}(-3+\pi+4\log
2-4\log(\pi T))+\cdots,$ (4.30)

where the infrared divergences will be exactly cancelled by the integral over
the infinitesimal circle. In order to obtain the contribution from the
infinitesimal circle, we turn to polar coordinate $\hat{z}=1/(\pi T)+\epsilon
e^{i\theta}$

$\displaystyle\int_{\mathcal{C}}d\hat{z}\frac{\gamma^{2}\pi^{3}T^{6}\hat{\omega}_{3}\hat{z}^{2}(1-3\pi^{2}T^{2}\hat{z}^{2})}{(1-\pi^{4}T^{4}\hat{z}^{4})^{2}}\log(1-\pi
T\hat{z})$ $\displaystyle=$ $\displaystyle\int_{-\pi}^{\pi}d\theta\,i\epsilon
e^{i\theta}\frac{\gamma^{2}\pi^{3}T^{6}\hat{\omega}_{3}\hat{z}^{2}(1-3\pi^{2}T^{2}\hat{z}^{2})}{(1-\pi^{4}T^{4}\hat{z}^{4})^{2}}\left[\log(\pi
T\epsilon)+i\pi+i\theta\right]\bigg{|}_{\hat{z}\to 1/(\pi T)+\epsilon
e^{i\theta}}$ $\displaystyle=$
$\displaystyle-\frac{i\hat{\omega}_{3}\gamma^{2}T^{2}}{4\epsilon}-\frac{i\hat{\omega}_{3}\gamma^{2}\pi
T^{3}}{2}\log\epsilon+\frac{1}{2}\gamma^{2}\hat{\omega}_{3}\pi^{2}T^{3}-\frac{1}{2}i\hat{\omega}_{3}\gamma^{2}\pi
T^{3}\log(\pi T)+\cdots.$ (4.31)

So, the result for $I_{3}$ is

$\displaystyle
I_{3}=\gamma^{2}\pi^{2}T^{3}\hat{\omega}_{3}\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{r}^{L}(\hat{\omega}_{3})-\frac{1}{8}i\hat{\omega}_{3}\gamma^{2}\pi
T^{3}(3\pi-2)\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{L}(\hat{\omega}_{3}).$
(4.32)

The integrals $I_{4},I_{5}$ could be computed in parallel with $I_{3}$. For
brevity, we just record the final results:

$\displaystyle I_{4}=-\frac{1}{8}i\hat{\omega}_{3}\pi
T^{3}(2+\pi)\hat{q}_{a}^{L}(\hat{\omega}_{3})\hat{q}_{a}^{T}(\hat{\omega}_{1})\hat{q}_{a}^{T}(\hat{\omega}_{2}),$
$\displaystyle
I_{5}=\pi^{2}T^{3}\hat{\omega}_{3}\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{T}(\hat{\omega}_{1})\hat{q}_{r}^{T}(\hat{\omega}_{3})-\frac{1}{4}i\hat{\omega}_{3}\pi
T^{3}(\pi-2)\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{T}(\hat{\omega}_{1})\hat{q}_{a}^{T}(\hat{\omega}_{3}).$
(4.33)

Finally, at the order $\mathcal{O}(\hat{\omega}^{2})$ we only consider
$arr$-terms in the cubic action. The integrals $I_{1},I_{2},I_{5}$ do not
contribute. The contributions from $I_{3,4,6,7}$ are simply computed via
residue theorem. The results are

$\displaystyle
I_{3}^{arr}=\gamma^{2}\pi^{2}T^{2}\hat{\omega}_{1}\hat{\omega}_{3}\hat{q}_{r}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{r}^{L}(\hat{\omega}_{3}),$
$\displaystyle
I_{4}^{arr}=\pi^{2}T^{2}\hat{\omega}_{1}\hat{\omega}_{3}\hat{q}_{r}^{T}(\hat{\omega}_{1})\hat{q}_{a}^{T}(\hat{\omega}_{2})\hat{q}_{r}^{L}(\hat{\omega}_{3}),$
$\displaystyle
I_{6}^{arr}=\frac{\gamma^{2}\pi^{2}T^{2}}{2}\hat{\omega}_{2}\hat{\omega}_{3}\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{r}^{L}(\hat{\omega}_{2})\hat{q}_{r}^{L}(\hat{\omega}_{3}),$
$\displaystyle
I_{7}^{arr}=\frac{\pi^{2}T^{2}}{2}\hat{\omega}_{2}\hat{\omega}_{3}\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{r}^{T}(\hat{\omega}_{2})\hat{q}_{r}^{T}(\hat{\omega}_{3}).$
(4.34)

Combining (4.25), (4.32), (4.33) and (4.34), we obtain the cubic order SK
effective action:

$\displaystyle S_{eff}^{(3)}=S_{eff}^{(3)LLL}+S_{eff}^{(3)LTT},$ (4.35)

where

$\displaystyle S_{eff}^{(3)LLL}=$
$\displaystyle\frac{\beta\gamma}{2\pi\alpha^{\prime}}\int\frac{d\hat{\omega}_{2}d\hat{\omega}_{3}}{(2\pi)^{2}}\left\\{\left[-\frac{3}{2}\gamma^{2}\pi^{2}T^{4}+\frac{1}{8}\gamma^{2}\pi(3\pi-2)T^{3}i\hat{\omega}_{3}\right]\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{L}(\hat{\omega}_{3})\right.$
$\displaystyle\left.-\frac{5}{2}\gamma^{2}\pi^{2}T^{3}\hat{\omega}_{3}\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{r}^{L}(\hat{\omega}_{3})-\frac{3}{2}\gamma^{2}\pi^{2}T^{2}\hat{\omega}_{2}\hat{\omega}_{3}\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{r}^{L}(\hat{\omega}_{2})\hat{q}_{r}^{L}(\hat{\omega}_{3})\right\\},$
$\displaystyle S_{eff}^{(3)LTT}=$
$\displaystyle\frac{\beta\gamma}{2\pi\alpha^{\prime}}\int\frac{d\hat{\omega}_{2}d\hat{\omega}_{3}}{(2\pi)^{2}}\left\\{\left[-\frac{3}{2}\pi^{2}T^{4}+\frac{1}{8}\pi(\pi+2)T^{3}i\hat{\omega}_{1}+\frac{1}{4}\pi(\pi-2)T^{3}i\hat{\omega}_{3}\right]\right.$
$\displaystyle\left.\times\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{T}(\hat{\omega}_{2})\hat{q}_{a}^{T}(\hat{\omega}_{3})-\frac{5}{2}\pi^{2}T^{3}\hat{\omega}_{3}\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{T}(\hat{\omega}_{2})\hat{q}_{r}^{T}(\hat{\omega}_{3})-\pi^{2}T^{2}\hat{\omega}_{1}\hat{\omega}_{3}\right.$
$\displaystyle\left.\times\hat{q}_{r}^{T}(\hat{\omega}_{1})\hat{q}_{a}^{T}(\hat{\omega}_{2})\hat{q}_{r}^{L}(\hat{\omega}_{3})-\frac{1}{2}\pi^{2}T^{2}\hat{\omega}_{2}\hat{\omega}_{3}\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{r}^{T}(\hat{\omega}_{2})\hat{q}_{r}^{T}(\hat{\omega}_{3})\right\\},$
(4.36)

where $\hat{\omega}_{1}+\hat{\omega}_{2}+\hat{\omega}_{3}=0$ is assumed.
Recall that at the order $\mathcal{O}(\hat{\omega}^{2})$ we only track
$arr$-type terms. (4.36) is the main result of this work.

The cubic order action (4.36) represents not only nonlinear interactions among
noises (the $aaa$-type terms) but also nonlinear interactions between
dynamical variable $\hat{q}_{r}$ and noise (the $aar$-type terms), which are
usually not covered in stochastic formulation, such as (1.1). Moreover, the
$aar$-type terms are analogous to multiplicative noise in classical stochastic
models [2]. The $arr$-type terms represents nonlinear $(\partial_{t}q)^{2}$
correction to the Langevin equation (1.1). However, presence of $aaa$-type
terms or $aar$-type terms makes it inconvenient to convert (via Legendre
transformation) the effective action (4.6) and (4.36) into a stochastic
equation for heavy quark position [17]. In fact, many problems, although
initially formulated as (generalized) Langevin equations with certain
stochastic forces, are indeed studied by converting into an action framework,
see [2]. It is thus interesting to explore physical consequences of these
terms by computing loop corrections to various observables, along the line of
[23, 25, 60, 61], which goes beyond the scope of present work.

Consistency check

$\bullet$ $Z_{2}$ reflection symmetry

Under the exchange of $\hat{q}_{1}^{L,T}\leftrightarrow\hat{q}_{2}^{L,T}$, the
SK effective action shall satisfy [17]:

$\displaystyle\left(S_{eff}[\hat{q}_{1}^{T},\hat{q}_{1}^{L};\hat{q}_{2}^{T},\hat{q}_{2}^{L}]\right)^{*}=-S_{eff}[\hat{q}_{2}^{T},\hat{q}_{2}^{L};\hat{q}_{1}^{T},\hat{q}_{1}^{L}],$
(4.37)

which is actually unitary requirement, representing a self-consistent
condition of the SK formalism. Under the constraint (4.37), in the effective
action (in time domain), the coefficient of any term with even numbers of
$a$-type fields should be purely imaginary, while the coefficient of a term
with odd numbers of $a$-type fields must be purely real. Obviously, our
results for quadratic and cubic actions pass this basic requirement.

$\bullet$ KMS constraints for three-point functions

From the perspective of string worldsheet, we work around a thermal state (an
emergent one on the worldsheet) with constant temperature, say (2.14) or
(2.15). As analyzed in appendix A, while the quadratic bulk action (2.21)
preserves worldsheet $\mathcal{T}$-symmetry, the cubic bulk action (2.22) does
explicitly break it. On the other hand, the KMS constraints (C.5) become
applicable only if the theory is $\mathcal{T}$-invariant [17]. Thus, it is
reasonable to expect that the cubic order results (4.36) will not satisfy
equilibrium KMS constraints (C.5), which we examine now.

Comparing (4.36) with (C.1)-(C.4), we can read off three-point correlators for
medium force $\hat{\mathcal{F}}$. Let us take the $LLL$-sector as an example
(see appendix C for nation convention):

$\displaystyle\frac{1}{3!}\hat{G}^{LLL}=-\frac{3\gamma^{2}\pi^{2}T^{4}}{2}+\frac{1}{8}i\hat{\omega}_{3}\gamma^{2}\pi
T^{3}(3\pi-2),$
$\displaystyle\frac{i}{2}\hat{H}_{1}^{LLL}=-\frac{5\gamma^{2}\pi^{2}T^{3}}{2}\hat{\omega}_{1},\quad\frac{i}{2}\hat{H}_{2}^{LLL}=-\frac{5\gamma^{2}\pi^{2}T^{3}}{2}\hat{\omega}_{2},\quad\frac{i}{2}\hat{H}_{3}^{LLL}=-\frac{5\gamma^{2}\pi^{2}T^{3}}{2}\hat{\omega}_{3},$
$\displaystyle\frac{1}{2}\hat{K}_{1}^{LLL}=-\frac{3\gamma^{2}\pi^{2}T^{2}}{2}\hat{\omega}_{2}\hat{\omega}_{3},\quad\frac{1}{2}\hat{K}_{2}^{LLL}=-\frac{3\gamma^{2}\pi^{2}T^{2}}{2}\hat{\omega}_{1}\hat{\omega}_{3},\quad\frac{1}{2}\hat{K}_{3}^{LLL}=-\frac{3\gamma^{2}\pi^{2}T^{2}}{2}\hat{\omega}_{1}\hat{\omega}_{2},$
(4.38)

where “hatted” $\hat{G},\hat{H},\hat{K}$ mean correlators of rescaled medium
force $\hat{\mathcal{F}}$. In (4.38), an overall factor
$\beta\gamma/(2\pi\alpha^{\prime})$ is ignored. Apparently, our results (4.38)
do not satisfy (C.5). The same conclusion holds for $LTT$-sector.

##  5 Summary and Outlook

In this work, we considered nonlinear corrections to stochastic dynamics of a
relativistic heavy quark in $\mathcal{N}=4$ SYM plasma, holographically
described by a trailing string in AdS5 black brane [30, 41, 42]. Based on the
holographic SK contour [45], we derived the SK effective action for the heavy
quark, up to cubic order in quark’s position.

At quadratic level, the SK effective action is computed first in the small
$\hat{\omega}$ limit and then for arbitrary value of $\hat{\omega}$, perfectly
satisfying “worldsheet” KMS condition, see (4.10). At cubic order, the SK
effective action is computed perturbatively in small $\hat{\omega}$ regime. We
observed that the KMS conditions for three-point functions of medium force are
no longer obeyed due to steady state motion of the heavy quark. This becomes
more transparent from string worldsheet’s viewpoint: while the state of a
trailing string results in an emergent static AdS black hole background on the
string worldsheet, it also renders the cubic order bulk action to break
worldsheet $\mathcal{T}$-invariance (see appendix A).

Another outcome of this work is on the technical side. We carried out a
preliminary study over the subtle issue arising from non-commutative feature
of the following two limits: the low frequency limit $\hat{\omega}/T\to 0$
versus $\epsilon\to 0$. Through brute-force calculations, we demonstrated that
this subtlety is accidently absent in the calculation of SK effective action
up to $\mathcal{O}(\hat{\omega}^{1})$. However, this would not hold beyond
first order in time derivative, and requires further investigation. We leave
this open question for future study.

The present work focused on steady state motion of a Brownian particle in a
static plasma medium. This can be extended in a few directions:

$\bullet$ First, it is interesting to consider a heavy quark moving in a
viscous neutral flow background [62, 63, 64, 65], which is dual to an open
string probing slowly-varying AdS black brane [66]. Such a study would be
helpful in understanding fluctuation-dissipation relations for a non-
equilibrium state [43, 67, 68].

$\bullet$ Second, the present study would be extended to more realistic
scenarios, say in the context of holographic models more closer to QCD plasma,
as considered recently in [69, 70, 59, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82].

$\bullet$ Last but not least, it is also interesting to consider spatially
extended objects, and particularly when the low energy dynamics involves
densities of conserved charges. One such example is the quantum critical
system driven by a strong electric field [83], aiming at revealing universal
scaling behaviors for out-of-equilibrium situation. A second interesting
instance is the steady state bubble nucleated within a chiral phase transition
[84]. It is worth exploring nonlinear stochastic effects in both examples via
holographic technique.

##  Appendix A Perturbative $S_{\rm NG}$ in Schwarzschild coordinate

In section 2.3, the perturbative Nambu-Goto actions (2.21) and (2.22) are
presented in the ingoing EF coordinate (2.15), which makes the properties
under $\mathcal{T}$-transformation obscure. In Schwarzschild coordinate
(2.14), the perturbative Nambu-Goto actions (2.21) and (2.22) become:

$\displaystyle S_{\rm NG}^{(2)}=$
$\displaystyle-\frac{1}{2\pi\alpha^{\prime}}\int
d\hat{t}\int_{0_{1}}^{0_{2}}\frac{d\hat{z}}{2\hat{z}^{2}}\left[-\frac{\gamma^{2}}{f(\hat{z})}(\partial_{\hat{t}}\hat{X}_{L})^{2}+\gamma^{2}f(\hat{z})(\partial_{\hat{z}}\hat{X}_{L})^{2}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.-\frac{1}{f(\hat{z})}(\partial_{\hat{t}}\hat{X}_{T})^{2}+f(\hat{z})(\partial_{\hat{z}}\hat{X}_{T})^{2}\right],$
(A.1) $\displaystyle S_{\rm NG}^{(3)}=$
$\displaystyle-\frac{\gamma\beta}{2\pi\alpha^{\prime}}\int
d\hat{t}\int_{0_{1}}^{0_{2}}\frac{d\hat{z}}{2\hat{z}^{2}f^{2}(\hat{z})}\bigg{\\{}-\gamma^{2}(\partial_{\hat{t}}\hat{X}_{L})^{3}-\gamma^{2}(\pi
T\hat{z})^{2}f(\hat{z})(\partial_{\hat{t}}\hat{X}_{L})^{2}\partial_{\hat{z}}\hat{X}_{L}$
$\displaystyle-\partial_{\hat{t}}\hat{X}_{L}(\partial_{\hat{t}}\hat{X}_{T})^{2}-2(\pi
T\hat{z})^{2}f(\hat{z})\partial_{\hat{t}}\hat{X}_{L}\partial_{\hat{t}}\hat{X}_{T}\partial_{\hat{z}}\hat{X}_{T}+\gamma^{2}f^{2}(\hat{z})\partial_{\hat{t}}\hat{X}_{L}(\partial_{\hat{z}}\hat{X}_{L})^{2}$
$\displaystyle-f^{2}(\hat{z})\partial_{\hat{t}}\hat{X}_{L}(\partial_{\hat{z}}\hat{X}_{T})^{2}+(\pi
T\hat{z})^{2}f(\hat{z})\partial_{\hat{z}}\hat{X}_{L}(\partial_{\hat{t}}\hat{X}_{T})^{2}+2f^{2}(\hat{z})\partial_{\hat{z}}\hat{X}_{L}\partial_{\hat{t}}\hat{X}_{T}\partial_{\hat{z}}\hat{X}_{T}$
$\displaystyle+\gamma^{2}(\pi
T\hat{z})^{2}f^{3}(\hat{z})(\partial_{\hat{z}}\hat{X}_{L})^{3}+(\pi
T\hat{z})^{2}f^{3}(\hat{z})\partial_{\hat{z}}\hat{X}_{L}(\partial_{\hat{z}}\hat{X}_{T})^{2}\bigg{\\}}.$
(A.2)

Now, from the worldsheet perspective, the $\mathcal{T}$-symmetry becomes more
transparent. The second order action (A.1) is invariant under the
$\mathcal{T}$-transformation:

$\displaystyle\mathcal{T}:\qquad\hat{t}\to-\hat{t},\qquad\hat{X}_{T,L}(\hat{z},\hat{t})\to\hat{X}_{T,L}(\hat{z},-\hat{t}).$
(A.3)

However, the cubic order action (A.2) is not invariant under
$\mathcal{T}$-transformation above. Moreover, it does not make sense to split
$S_{\rm NG}^{(3)}$ into $\mathcal{T}$-invariant part and
$\mathcal{T}$-breaking part, since it is the whole object $S_{\rm NG}^{(3)}$
that is a Lorentzian scalar. In some sense, the $\mathcal{T}$-breaking action
$S_{\rm NG}^{(3)}$ is a reflection of steady state motion for the quark.

##  Appendix B Subtlety due to non-commutativity of $\epsilon\to 0$ versus
$\hat{\omega}\to 0$

In this appendix, we demonstrate that the non-commutativity issue of
$\epsilon\to 0$ versus $\hat{\omega}\to 0$ becomes accidentally absent for
present study. This validates the treatment of subsection 4.2.

Recall that each integrand in (4.19) potentially has an oscillating factor
$(1-\pi T\hat{z})^{i\hat{\omega}/(2\pi T)}$, which results in the non-
commutativity issue (4.20). With the expression (4.23), the factor
$e^{i\hat{\omega}\chi(\hat{z})}$ can be rewritten as a product of an
oscillating part and a regular part:

$\displaystyle e^{i\hat{\omega}\chi(\hat{z})}=(1-\pi
T\hat{z})^{i\hat{\omega}/(2\pi T)}{\rm exp}\left\\{-\frac{i\hat{\omega}}{\pi
T}\arctan(\pi T\hat{z})-\frac{i\hat{\omega}}{2\pi T}\log(1+\pi
T\hat{z})\right\\},$ (B.1)

where the singular portion $(1-\pi T\hat{z})^{i\hat{\omega}/(2\pi T)}$ will
not be expanded in small $\hat{\omega}$ limit before $\epsilon\to 0$. Once the
integrals (4.19) are done, we extract the low frequency limits.

We find it more convenient to rewrite $B$ of (3.24) as

$\displaystyle
B(\hat{z},\hat{\omega})=B_{1}(\hat{z},\hat{\omega})+B_{2}(\hat{z},\hat{\omega})=B_{1}(\hat{z},\hat{\omega})+(1-\pi
T\hat{z})^{i\hat{\omega}/(2\pi T)}C(\hat{z},\hat{\omega}),$ (B.2)

where $B_{1},C$ are regular functions over the domain enclosed by the contour

$\displaystyle
B_{1}(\hat{z},\hat{\omega})=\frac{1}{2}\coth\frac{\beta_{0}\hat{\omega}}{2}\frac{\Phi^{\rm
ig}(\hat{z},\hat{\omega})}{\Phi^{\rm ig(0)}(\hat{\omega})},$ $\displaystyle
C(\hat{z},\hat{\omega})=-\frac{1}{1-e^{-\beta_{0}\hat{\omega}}}\frac{\Phi^{\rm
ig}(\hat{z},-\hat{\omega})}{\Phi^{\rm ig(0)}(-\hat{\omega})}{\rm
exp}\left\\{-\frac{i\hat{\omega}}{\pi T}\arctan(\pi
T\hat{z})-\frac{i\hat{\omega}}{2\pi T}\log(1+\pi T\hat{z})\right\\}.$ (B.3)

While the oscillating factor $(1-\pi T\hat{z})^{i\hat{\omega}/(2\pi T)}$ is
kept unexpanded, it is always legitimate to expand all regular parts, such as
$A,B_{1},C$, in small $\hat{\omega}$. In the small $\hat{\omega}$ limit, the
functions $A,B_{1},C$ and their radial derivatives scale as

$\displaystyle A=\mathcal{O}(\hat{\omega}^{0})+\cdots,\qquad\quad
B_{1}=\mathcal{O}(\hat{\omega}^{-1})+\cdots,\qquad\quad
C=\mathcal{O}(\hat{\omega}^{-1})+\cdots,$
$\displaystyle\partial_{\hat{z}}A=\mathcal{O}(\hat{\omega}^{1})+\cdots,\qquad\partial_{\hat{z}}B_{1}=\mathcal{O}(\hat{\omega}^{0})+\cdots,\qquad\partial_{\hat{z}}C=\mathcal{O}(\hat{\omega}^{0})+\cdots.$
(B.4)

Before calculating contour integrals (4.19), we make a formal analysis about
each integrand. Recall that $A,B_{1}$ are regular, and $B_{2}$ has one branch
cut at the horizon. So, in order to have non-vanishing contribution from each
contour integral $I_{i}$ in (4.19), the integrand must contain one
$B_{2}$-factor at least. Moreover, since
$\partial_{\hat{z}}A\sim\mathcal{O}(\hat{\omega}^{1})$, a term containing a
factor $\partial_{\hat{z}}A\partial_{\hat{z}}A$ might be of order
$\mathcal{O}(\hat{\omega}^{2})$.

We take $I_{3}$ as an example to illustrate the computation. In terms of
$A,B_{1},B_{2}$, the integral $I_{3}$ becomes,

$\displaystyle I_{3}=\int_{0_{1}}^{0_{2}}d\hat{z}$
$\displaystyle\frac{-i\hat{\omega}_{3}\gamma^{2}}{2\hat{z}^{2}}(1-3\pi^{2}T^{2}\hat{z}^{2})\left[2\partial_{\hat{z}}B_{2}(\hat{z},\hat{\omega}_{1})\partial_{\hat{z}}A(\hat{z},\hat{\omega}_{2})\,A(\hat{z},\hat{\omega}_{3})\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{r}^{L}(\hat{\omega}_{2})\hat{q}_{r}^{L}(\hat{\omega}_{3})\right.$
$\displaystyle\qquad\qquad\left.+2\partial_{\hat{z}}B_{2}(\hat{z},\hat{\omega}_{1})\partial_{\hat{z}}A(\hat{z},\hat{\omega}_{2})B_{1}(\hat{z},\hat{\omega}_{3})\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{r}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{L}(\hat{\omega}_{3})\right.$
$\displaystyle\qquad\qquad\left.+2\partial_{\hat{z}}B_{2}(\hat{z},\hat{\omega}_{1})\partial_{\hat{z}}B_{1}(\hat{z},\hat{\omega}_{2})A(\hat{z},\hat{\omega}_{3})\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{r}^{L}(\hat{\omega}_{3})\right.$
$\displaystyle\qquad\qquad\left.+2\partial_{\hat{z}}B_{2}(\hat{z},\hat{\omega}_{1})\partial_{\hat{z}}B_{1}(\hat{z},\hat{\omega}_{2})B_{1}(\hat{z},\hat{\omega}_{3})\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{L}(\hat{\omega}_{3})\right.$
$\displaystyle\qquad\qquad\left.+\partial_{z}A(\hat{z},\hat{\omega}_{1})\partial_{z}A(\hat{z},\hat{\omega}_{2})B_{2}(\hat{z},\hat{\omega}_{3})\hat{q}_{r}^{L}(\hat{\omega}_{1})\hat{q}_{r}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{L}(\hat{\omega}_{3})\right.$
$\displaystyle\qquad\qquad\left.+\partial_{\hat{z}}B_{1}(\hat{z},\hat{\omega}_{1})\partial_{\hat{z}}B_{1}(\hat{z},\hat{\omega}_{2})B_{2}(\hat{z},\hat{\omega}_{3})\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{L}(\hat{\omega}_{3})\right.$
$\displaystyle\qquad\qquad\left.+2\partial_{\hat{z}}A(\hat{z},\hat{\omega}_{1})\partial_{\hat{z}}B_{1}(\hat{z},\hat{\omega}_{2})B_{2}(\hat{z},\hat{\omega}_{3})\hat{q}_{r}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{L}(\hat{\omega}_{3})\right.$
$\displaystyle\qquad\qquad\left.+\partial_{\hat{z}}B_{2}(\hat{z},\hat{\omega}_{1})\partial_{\hat{z}}B_{2}(\hat{z},\hat{\omega}_{2})A(\hat{z},\hat{\omega}_{3})\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{r}^{L}(\hat{\omega}_{3})\right.$
$\displaystyle\qquad\qquad\left.+\partial_{\hat{z}}B_{2}(\hat{z},\hat{\omega}_{1})\partial_{\hat{z}}B_{2}(\hat{z},\hat{\omega}_{2})B_{1}(\hat{z},\hat{\omega}_{3})\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{L}(\hat{\omega}_{3})\right.$
$\displaystyle\qquad\qquad\left.+2\partial_{\hat{z}}B_{2}(\hat{z},\hat{\omega}_{1})\partial_{\hat{z}}A(\hat{z},\hat{\omega}_{2})B_{2}(\hat{z},\hat{\omega}_{3})\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{r}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{L}(\hat{\omega}_{3})\right.$
$\displaystyle\qquad\qquad\left.+2\partial_{\hat{z}}B_{2}(\hat{z},\hat{\omega}_{1})\partial_{\hat{z}}B_{1}(\hat{z},\hat{\omega}_{2})B_{2}(\hat{z},\hat{\omega}_{3})\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{L}(\hat{\omega}_{3})\right.$
$\displaystyle\qquad\qquad\left.+\partial_{\hat{z}}B_{2}(\hat{z},\hat{\omega}_{1})\partial_{\hat{z}}B_{2}(\hat{z},\hat{\omega}_{2})\partial_{\hat{z}}B_{2}(\hat{z},\hat{\omega}_{3})\hat{q}_{a}^{L}(\hat{\omega}_{1})\hat{q}_{a}^{L}(\hat{\omega}_{2})\hat{q}_{a}^{L}(\hat{\omega}_{3})\right].$
(B.5)

Thanks to the delta function
$\delta(\hat{\omega}_{1}+\hat{\omega}_{2}+\hat{\omega}_{3})$, the last term of
(B.5) does not contain the singular factor $(1-\pi
T\hat{z})^{i\hat{\omega}/(2\pi T)}$, and will be computed via the residue
theorem. The rest terms in (B.5) contain an overall singular factor $(1-\pi
T\hat{z})^{i\hat{\omega}/(2\pi T)}$, and could be schematically written as

$\displaystyle\mathcal{I}=\int_{0_{1}}^{0_{2}}\frac{d\hat{z}}{\hat{z}^{2}}(1-\pi
T\hat{z})^{\pm i\hat{\omega}/(2\pi T)}\mathcal{H}(\hat{z},\hat{\omega}),$
(B.6)

where $\mathcal{H}(\hat{z},\hat{\omega})$ contains simple pole and/or second-
order pole near the horizon. Note that from (B.5) to (B.6), we ignored purely
numerical factor as well as potential powers of frequency
$\hat{\omega}^{m}(m=1~{}{\rm or}~{}2)$. Therefore, for the purpose of
evaluating $I_{3}$ up to $\mathcal{O}(\hat{\omega}^{1})$, it is sufficient to
compute (B.6) up to $\mathcal{O}(\hat{\omega}^{1})$. For a generic function
$\mathcal{H}(\hat{z},\hat{\omega})$ it is challenging to obtain analytical
results for (B.6). Consider Laurent expansion of
$\mathcal{H}(\hat{z},\hat{\omega})$ near the horizon

$\displaystyle\mathcal{H}(\hat{z},\hat{\omega})=\sum_{n=-2}^{\infty}\mathcal{H}_{n}(\hat{\omega})(1-\pi
T\hat{z})^{n},$ (B.7)

which has a convergence radius larger than $1/(\pi T)$. Therefore, in the
domain enclosed by the radial contour, it is legal to represent
$\mathcal{H}(\hat{z},\hat{\omega})$ by its Laurent expansion (B.7). Here, in
the small frequency limit $\mathcal{H}_{n}(\hat{\omega})$ is a Taylor series
of frequency. So, the original task of computing (B.6) now boils down to
calculating

$\displaystyle\mathcal{I}_{n}=\int_{0_{1}}^{0_{2}}\frac{d\hat{z}}{\hat{z}^{2}}(1-\pi
T\hat{z})^{\pm i\hat{\omega}/(2\pi T)}(1-\pi T\hat{z})^{n},\qquad
n=-2,-1,0,1,\cdots,$ (B.8)

which could be worked out analytically for generic $\hat{\omega}$. Afterwards,
we extract the small frequency behavior of $\mathcal{I}_{n}$

$\displaystyle\mathcal{I}_{n}=\mp\hat{\omega}\frac{{}_{2}F_{1}(2,n+1;n+2;1-\Lambda)}{(n+1)T}+\mathcal{O}(\hat{\omega}^{2}),\qquad\quad\,n\geq
1,$ $\displaystyle\mathcal{I}_{n}=\mp\frac{\hat{\omega}}{\Lambda
T}+\mathcal{O}(\hat{\omega}^{2})\qquad\qquad\qquad\qquad\qquad\qquad\qquad
n=0,$ $\displaystyle\mathcal{I}_{n}=2i\pi\mp\frac{\hat{\omega}}{\Lambda
T}+\mathcal{O}(\hat{\omega}^{2}),\qquad\qquad\qquad\qquad\qquad\quad~{}~{}n=-1,$
$\displaystyle\mathcal{I}_{n}=4i\pi+\mathcal{O}(\hat{\omega}),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad
n=-2,$ (B.9)

where $\Lambda$ represents a UV cutoff near the AdS boundary. It is
straightforward to check that the results (B.9) could be correctly obtained by
first expanding the integrand of (B.8) in small $\hat{\omega}$ and then doing
the radial integral. However, this latter approach cannot correctly cover
higher order terms omitted in (B.9), which correspond to higher order terms in
the SK action.

Now, we turn to the $arr$-terms at the order $\mathcal{O}(\hat{\omega}^{2})$,
which correspond to the first line and fifth line in (B.5). With the scaling
(B.4), it is obvious that only the first term of (B.5) is relevant. Moreover,
only the lowest order results in (B.9) are required.

The main lesson from analysis above is as follows. In order to obtain SK
effective action to the order $\mathcal{O}(\hat{\omega}^{1})$, it is legal
although accidentally to first expand the integrands (including $(1-\pi
T\hat{z})^{\pm i\hat{\omega}/(2\pi T)}$) in (4.19) in small $\hat{\omega}$,
and then implement the radial integral. While generically this might not be
true for higher order terms, it does work for extracting $arr$-terms at the
order $\mathcal{O}(\hat{\omega}^{2})$.

##  Appendix C KMS conditions for three-point functions

Consider a quantum many-body system in a thermal state, which also preserves
$\mathcal{PT}$-symmetry. The KMS condition, satisfied by the generating
functional $W$, sets constraints among three-point functions [57]. These
constraints were also re-derived and extended to situation with multiple
variables in [17]. In this appendix, we review these constraints for
completeness. Here, we closely follow subsection II D of [17].

At cubic level, the generating functional can be written as

$\displaystyle W_{\rm
cub}=i\left[\frac{1}{3!}G^{lmn}\phi_{a}^{l}\phi_{a}^{m}\phi_{a}^{n}+\frac{i}{2}H^{lmn}\phi_{a}^{l}\phi_{a}^{m}\phi_{r}^{n}+\frac{1}{2}K^{lmn}\phi_{a}^{l}\phi_{r}^{m}\phi_{r}^{n}\right],$
(C.1)

where the indices $l$ etc labels different bosonic Hermitian operators
$\mathcal{O}^{l}$ and associated sources $\phi^{l}$. Each term in (C.1) should
be understood as a convolution in Fourier space, e.g.,

$\displaystyle
G^{lmn}\phi_{a}^{l}\phi_{a}^{m}\phi_{a}^{n}=\int\frac{d\omega_{2}d\omega_{3}}{(2\pi)^{2}}G^{lmn}(\omega_{1},\omega_{2},\omega_{3})\phi_{a}^{l}(\omega_{1})\phi_{a}^{m}(\omega_{2})\phi_{a}^{n}(\omega_{3}),\quad\omega_{1}+\omega_{2}+\omega_{3}=0,$
(C.2)

where spatial momenta are suppressed. The three-point functions
$G_{\alpha_{1}\alpha_{2}\alpha_{3}}^{lmn}$, with $\alpha_{1,2,3}=r~{}{\rm
or}~{}a$, are third order functional derivatives of $W_{\rm cub}$ with respect
to sources $\phi_{r,a}^{l}$. Then, we have

$\displaystyle G^{lmn}=-G_{rrr}^{lmn},\qquad H^{lmn}=G_{rra}^{lmn},\qquad
K^{lmn}=G_{raa}^{lmn}.$ (C.3)

To present KMS constraints for three-point functions, it is convenient to
introduce the following notations:

$\displaystyle H_{3}^{lmn}\equiv G_{rra}^{lmn},\qquad H_{2}^{lmn}\equiv
G_{rar}^{lmn},\qquad H_{1}^{lmn}\equiv G_{arr}^{lmn},$ $\displaystyle
K_{1}^{lmn}\equiv G_{raa}^{lmn},\qquad K_{2}^{lmn}\equiv G_{ara}^{lmn},\qquad
K_{3}^{lmn}\equiv G_{aar}^{lmn}.$ (C.4)

Note that in (C.3) and (C.4) variables $\omega$’s are further suppressed. For
a $\mathcal{PT}$-invariant theory, KMS condition applied to three-point
functions implies

$\displaystyle
H_{1}=\frac{i}{2}(N_{3}+N_{2})K_{1}^{*}-\frac{i}{2}(N_{2}K_{3}+N_{3}K_{2}),$
$\displaystyle
H_{2}=\frac{i}{2}(N_{3}+N_{1})K_{2}^{*}-\frac{i}{2}(N_{1}K_{3}+N_{3}K_{1}),$
$\displaystyle
H_{3}=\frac{i}{2}(N_{1}+N_{2})K_{3}^{*}-\frac{i}{2}(N_{1}K_{2}+N_{2}K_{1}),$
$\displaystyle
G=\frac{1}{4}\left[(K_{1}^{*}+K_{2}^{*}+K_{3}^{*})+2N_{2}N_{3}{\rm
Re}K_{1}+2N_{1}N_{3}{\rm Re}K_{2}+2N_{1}N_{2}{\rm Re}K_{3}\right],$ (C.5)

where for brevity the indices $l,m,n$ in the functions (C.3) and (C.4) are
suppressed. Moreover, we introduced

$\displaystyle N_{a}\equiv\coth\frac{\beta_{0}\omega_{a}}{2},\qquad a=1,2,3.$
(C.6)

The first three equations in (C.5) satisfy permutation symmetry among indices
$(123)$.

In order to guarantee regularity in the limit $\omega\to 0$, the first three
of (C.5) will involve some terms in (C.1) with at least one time-derivative,
while the last of (C.5) will involve some terms in (C.1) with at least two
time-derivatives. This explains why we need to additionally track $arr$-type
terms in (4.36) at $\mathcal{O}(\hat{\omega}^{2})$.

## Acknowledgements

This work was supported by the Natural Science Foundation of China (NSFC)
under the grant No. 11705037.

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