11institutetext: University of Southampton, University Road, Southampton SO17
1BJ, United Kingdom22institutetext: International Centre for Theoretical
Sciences (ICTS-TIFR), Shivakote, Hesaraghatta Hobli, Bengaluru 560089, India.

# Open effective theory of scalar field in rotating plasma

Bidisha Chakrabarty 2 Aswin P M<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the effective dynamics of an open scalar field interacting with a
strongly-coupled two-dimensional rotating CFT plasma. The effective theory is
determined by the real-time correlation functions of the thermal plasma. We
employ holographic Schwinger-Keldysh path integral techniques to compute the
effective theory. The quadratic effective theory computed using holography
leads to the linear Langevin dynamics with rotation. The noise and dissipation
terms in this equation get related by the fluctuation-dissipation relation in
presence of chemical potential due to angular momentum. We further compute
higher order terms in the effective theory of the open scalar field. At
quartic order, we explicitly compute the coefficient functions that appear in
front of various terms in the effective action in the limit when the
background plasma is slowly rotating. The higher order effective theory has a
description in terms of the non-linear Langevin equation with non-Gaussianity
in the thermal noise.

##  1 Introduction

Real time correlation functions are useful objects to study in quantum field
theories. For a field theory in vaccum or thermal state, the Lorentzian
correlators can be computed by analytically continuing the Euclidean
correlation functions. However for strongly coupled field theories, a direct
computation of correlation functions are extremely challenging. For
holographic strongly coupled conformal field theories one relies on the
celebrated anti-de-sitter/conformal field theory (AdS/CFT) correspondence
Maldacena:1997re  to compute CFT correlation functions using holography. The
duality in its original form was proposed to calculate Euclidean correlation
functions in strongly coupled $d$ dimensional CFTs by doing computations on
the weakly coupled $d+1$ dimensional AdS gravity side Gubser:1998bc ;
Witten:1998qj . This recipe was further extended to compute Lorentzian
correlation functions of the CFT using holography in Son:2002sd . One of the
main ingredients in this extension was to impose ingoing wave boundary
condition at the horizon of AdS black holes to get the Lorentzian correlation
functions of the thermal CFT. The authors of Son:2002sd  argued that there
would be no boundary contribution from the horizon in the calculation of
retarded bulk to boundary Green’s function. The prescription was further
developed for the maximally extended Kruskal spacetime by computing all bulk
to boundary propagators for a two sided black hole Herzog:2002pc . The
corresponding CFT path integral contour corresponds to the finite temperature
Schwinger-Keldysh (SK) contour with doubled degrees of freedom.

The above prescription was extensively used for the computation of real time
two point functions of the CFT using holography in various scenarios
Barnes:2010jp ; Son:2009vu ; CaronHuot:2011dr ; Chesler:2011ds ; Botta-
Cantcheff:2018brv ; Botta-Cantcheff:2019apr ; de2009brownian . However a
viable prescription for computing real time $n$-point functions required a
detailed description of the bulk manifold over which the path integral has to
be performed to compute strongly coupled CFT correlation functions
holographically. A proposal of such a bulk path integral framework was given
in Skenderis:2008dh ; Skenderis:2008dg . Since the bulk path integral contour
asymptotes to the CFT Schwinger-Keldysh contour at the boundary, the idea is
to fill in the imaginary and real time segments of the CFT contour by
Euclidean and Lorentzian geometries in the bulk. One has to impose appropriate
matching conditions (fields and their conjugate momenta have to be continuous)
at the corners of the contour where the metric changes signature. At finite
temperature, the bulk manifold in this prescription has two copies of half
Euclidean and half Lorentzian $AdS$ black brane geometries (glued across the
initial time slice) and the two copies are then stitched together across a
spacelike hypersurface. The corresponding CFT state is the thermofield double
(an entangled pure state). Given the prescription of entire path integration
contour on the gravity side, the generating functional of the CFT correlation
functions can be obtained by integrating over the bulk fields with sources as
boundary conditions where the operators inside the CFT correlator couple to
the sources.

Recently Glorioso, Crossley, Liu in Glorioso:2018mmw  have given a proposal on
how to handle the near horizon region of the holographic contour, that is, how
to stitch the two copies of the black branes across their horizons. This is a
crucial point to address since the near horizon regions are potential sources
of IR divergences once interactions are turned on atmaja2014holographic . The
proposal of Glorioso:2018mmw  suggests that the bulk dual of the asymptotic
Schwinger-Keldysh contour is given by a complexified double-sheeted spacetime.
The two copies of the black branes in the bulk are stitched by a ‘horizon cap’
across their future horizons. The CFT state in this case is not the
thermofield double, since in the bulk the prescription involves spacetime only
outside the future horizon. In Glorioso:2018mmw  the authors have computed the
quadratic effective actions of probe scalar and gauge fields to test their
formalism (also see deBoer:2018qqm ).

In Chakrabarty:2019aeu  the authors use the ‘horizon cap’ boundary conditions
of Glorioso:2018mmw  to calculate the quartic order effective action of a
probe quark moving in a strongly coupled thermal CFT plasma (bath) using
holography. This gives rise to the leading non-linear corrections to the
Brownian motion of the heavy quark. If the CFT bath is sufficiently forgetful
(that is when the bath correlators decay exponentially fast in time), the
effective theory of the quark becomes local in time (Markovian regime). The
local effective theory of the quark precisely matches the effective theory
obtained in Chakrabarty:2019qcp  (see also Chakrabarty:2018dov ).

Effective theories of a quantum Brownian particle (open probe) interacting
with various baths were obtained by integrating out the bath degrees of
freedom Feynman:1963fq ; Caldeira:1982iu ; Caldeira:1982uj ; Breuer:2002pc .
When the Brownian particle is linearly coupled to the bath degrees of freedom,
particle’s effective theory becomes quadratic. This problem has a classical
stochastic description in terms of the linear Langevin equation with a
Gaussian noise. In thermal equilibrium noise and dissipation terms in the
Langevin equation are related by the fluctuation dissipation relation. In
Chakrabarty:2018dov  the most general form of the local effective theory upto
cubic order was constructed where the authors considered a quantum Brownian
particle (open system) weakly interacting with a large thermal bath (made of
two sets of harmonic oscillators) via cubic interaction. This can be seen as a
perturbation over the well-known Caldeira-Leggett model. The effective theory
in Chakrabarty:2018dov  is described by a non-linear Langevin equation with a
non-gaussian noise. If the bath has microscopic time-reversal invariance then
the non-Gaussianity in the thermal noise gets related to the thermal jitter in
the damping constant of the Brownian particle. This is the generalisation of
the fluctuation dissipation relation. The path integral in the stochastic
problem is related to an underlying Schwinger-Keldysh quantum path integral.
In Chakrabarty:2019qcp  the authors extended their work by constructing an
effective theory of the particle upto quartic order.

In Chakrabarty:2019aeu  the validity of the effective theory is tested for a
strongly coupled bath using holography. Based on these analysis, in
Jana:2020vyx  the authors study open quantum field theories using holographic
Schwinger-Keldysh path integral. The authors consider an interacting scalar
quantum field theory (the system) coupled to a holographic field theory in $d$
spacetime dimensions at finite temperature (the environment). They study the
effects of integrating out the environment and obtain the effective dynamics
of the resulting open quantum field theory in a derivative expansion in small
frequency and momentum. For discussions on open quantum field theories and
their features see Avinash:2017asn ; Avinash:2019qga . In 1830227  the authors
study fermionic open effective field theories coupled to holographic baths.
They show how the holographic Schwinger-Keldysh path integral framework leads
to boundary correlators that automatically satisfy fermionic Kubo, Martin,
Schwinger (KMS) relations Kubo:1957mj ; Kubo_1966 . Computing the bulk on-
shell action they obtain the influence phase (the effect of integrating out
the bath degrees of freedom on the probe action) of the probe fermion in a
derivative expansion. In 1830471  the authors extend this analysis to charged
fermions probing Reissner-Nordström black hole background.

In this paper we extend the analysis of Jana:2020vyx  to rotating BTZ black
hole. We will consider an interacting probe scalar quantum field theory (the
system) coupled to a strongly coupled two dimensional rotating CFT plasma
(bath) at finite temperature and chemical potential.111In Atmaja:2012jg ;
atmaja2014holographic  the author studied holographic Brownian motion (linear
Langevin dynamics) of a heavy quark in a two dimensional rotating plasma. In
the bulk the probe scalar field is coupled to a rotating BTZ black hole
Banados:1992wn ; carlip19952+ . We consider a natural extension of the
holographic Schwinger-Keldysh path integral framework of Glorioso:2018mmw  to
the rotating BTZ black hole with two horizons. We study the effects coming
from integrating out the bath degrees of freedom on the effective action of
the open scalar field theory. The open scalar field gets dragged due to the
rotation of the background thermal plasma and is described by a generalised
nonlinear Langevin equation.

Following is the outline of the paper. We briefly review the gravitational
Schwinger-Keldysh path integral framework of Glorioso:2018mmw  and its natural
extension to the rotating BTZ black hole in section 2. In section 3 we study
the dynamics of a massive probe scalar field $\Phi$ in BTZ background. We
solve the scalar wave equation with appropriate boundary conditions to find
the ingoing and outgoing bulk to boundary green’s functions. We compute the
influence functionals (quadratic and higher order) of the open scalar field
holographically in section 4.222In Son:2002sd , the authors computed quadratic
effective theory and two point functions of the scalar field. In presence of
rotation, we study the stochastic dynamics governed by a non-linear Langevin
equation resulting from the open effective field theory in section 5. We also
find the generalised fluctuation dissipation relations obeyed by the
parameters of the non-linear Langevin equation. Finally we summarise our
results and discuss possible future directions in section 6.

Various technical details of the computation are relegated to the appendices.
In appendix A we provide details of the computation of quadratic influence
functional of the open scalar field in ‘retarded-advanced’ (RA) basis. In
appendix B we discuss the massive scalar field solution in a derivative
expansion in small frequency and momentum. The derivative expansion of the
solution is particularly useful in computing higher order effective theory of
the open scalar field.

##  2 Holographic SK prescription

Before we start reviewing the holographic SK path integral framework of
Glorioso:2018mmw , let us describe the path integral contour in the boundary
CFT on which the CFT correlation functions are defined. For the system
prepared in an initial thermal state, the Schwinger-Keldysh time contour as
shown in Fig 1 has imaginary (vertical leg) as well as real legs (horizontal
lines). The initial and final points of the contour have to be identified on a
thermal circle with periodicity $\beta$, where $\beta$ is the inverse
temperature of the CFT.

$-i\epsilon$$-i(\beta-\epsilon)$ Figure 1: Schwinger-Keldysh imaginary time
contour with an initial thermal state.

In Glorioso:2018mmw  the authors propose to fill in this boundary time contour
by a complexified doubled bulk space-time as shown in Fig 2.

$\text{Re}\ (r)$$\text{Im}\
(r)$$\times$$<$${\scriptsize{M_{L}}}$$>$$M_{R}$$r_{h}$$\qquad\qquad
r=r_{c}+i\varepsilon$$\qquad\qquad r=r_{c}-i\varepsilon$ Figure 2: Picture of
bulk at a fixed $v$: red line is a branch cut from horizon to boundary

Consider the Schwarzschild AdSd+1 black brane written in ingoing Eddington-
Finkelstein coordinates that are regular at the future horizon,

$ds^{2}=-r^{2}\,\left(1-\frac{r_{h}^{d}}{r^{d}}\right)\,dv^{2}+2\,dv\,dr+r^{2}\,d{\bf
x}^{2}\,.$ (1)

The coordinate $v$ becomes the time coordinate $t$ on the boundary of the
spacetime at $r\to\infty$. One has to consider two copies of the black brane
in the bulk that have to be stitched in a certain way across their future
horizon Glorioso:2018mmw . The proposed bulk spacetime correctly asymptotes to
the aforementioned Schwinger-Keldysh time contour. In the bulk the radial
coordinate has to be complexified to handle the near horizon region as
proposed in Glorioso:2018mmw . Define a complex radial tortoise coordinate
$\zeta$

$\frac{dr}{d\zeta}=\frac{i\,\beta}{2}\,r^{2}\,\left(1-\frac{r_{h}^{d}}{r^{d}}\right)\,,$
(2)

where $\beta=\frac{4\pi}{d\,r_{h}}$ is also the inverse Hawking temperature of
the black hole. $\zeta$ has a logarithmic branch point at $r=r_{h}$ coming
from the integral of the blackening factor (about its zero). The branch cut in
$\zeta$ is taken from $r=r_{h}$ to $r=\infty$. The jump of real part of
$\zeta$ across the horizon is 1. Hence $\zeta$ parametrizes a double-sheeted
spacetime that describes the bulk extension of the boundary Schwinger-Keldysh
contour. On each sheet imaginary part of $\zeta$ runs from $0$ at the AdS
boundary to $\infty$ at the horizon. The real part of $\zeta$ changes between
the two sheets and is given by the monodromy around the horizon. The real part
of $\zeta$ is chosen to be zero on the left sheet $M_{L}$, then the real part
of $\zeta$ becomes one on the right sheet $M_{R}$ after picking up the
monodromy at the horizon. We will also have a UV cut-off of the geometry at
$r=r_{c}$ for computational purposes. Hence asymptotically

$\zeta(r_{c}+i\,\varepsilon)=0\,,\qquad\zeta(r_{c}-i\,\varepsilon)=1\,.$ (3)

The metric written in $\zeta$ coordinate becomes

$ds^{2}=-r^{2}\,\left(1-\frac{r_{h}^{d}}{r^{d}}\right)\,dv^{2}+i\,\beta\,r^{2}\,\left(1-\frac{r_{h}^{d}}{r^{d}}\right)\,dv\,d\zeta+r^{2}\,d{\bf
x}^{2}\,,$ (4)

where $r(\zeta)$ is obtained by integrating (2). The complex tortoise
coordinate for Schwarzschild-AdSd+1 geometry becomes Jana:2020vyx

$\zeta+\zeta_{c}=\frac{i\,d}{2\pi\,(d-1)}\left(\frac{r}{r_{h}}\right)^{d-1}\,{}_{2}F_{1}\left(1,\frac{d-1}{d};2-\frac{1}{d};\frac{r^{d}}{r_{h}^{d}}\right),$
(5)

where $\zeta_{c}$ is considered to make $\zeta=0$ at $r=r_{c}+i\,\varepsilon$.
The branch-cut of the hypergeometric function runs from $r=r_{h}$ to
$r=\infty$. With this set up, one has to consider all bulk fields to live on a
complex $\zeta$ space and think of the classical bulk action to be a contour
integral over the complex tortoise coordinate $\zeta$. Hence

$S_{\text{bulk}}=\oint d\zeta\int
d^{d}x\,\sqrt{-g}\;\mathcal{L}[g_{AB},\Phi]\,$ (6)

where $x^{\mu}$ are the boundary coordinates. This action serves as the
starting point for computations of influence functionals.

In the following we extend the holographic contour mentioned above for
rotating BTZ black holes with outer and inner horizons. The two horizons
correspond to two branch points of the complexified tortoise coordinate in
this case. Let us start by writing the rotating BTZ black hole metric in
standard coordinates Son:2002sd

$\displaystyle\mathrm{d}s^{2}=-f\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{f}+r^{2}\bigl{(}\mathrm{d}\phi-
n_{\phi}\,\mathrm{d}t\bigr{)}^{2}\,;\
f(r)=\frac{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}{\,r^{2}}\,,\
n_{\phi}(r)=\frac{r_{+}r_{-}}{\,r^{2}}\,$ (7)

where $n_{\phi}(r)$ is the angular velocity of the black hole at radius r. We
will denote the two angular velocities at the two horizons to be
$n_{\phi}(r_{+})=\frac{r_{-}}{r_{+}}=\mu_{+}$ and
$n_{\phi}(r_{-})=\frac{r_{+}}{r_{-}}=\mu_{-}$. The two inverse temperatures
associated to the two horizons are denoted by $\beta_{+}$ and $\beta_{-}$
where

$\displaystyle\beta_{\pm}=\frac{2\pi r_{\pm}}{r_{+}^{2}-r_{-}^{2}}\ .$ (8)

The surface gravities $\kappa_{\pm}$ of the two horizons are

$\displaystyle\kappa_{\pm}=\frac{r_{+}^{2}-r_{-}^{2}}{r_{\pm}}\ .$ (9)

Let us now introduce the ingoing Eddington-Finkelstein coordinates for the
rotating BTZ metric that are regular at the future horizon

$\displaystyle\mathrm{d}v=\mathrm{d}t+\frac{\mathrm{d}r}{f}\ ,\
\mathrm{d}\phi^{\prime}=\mathrm{d}\phi+\frac{n_{\phi}}{f}\mathrm{d}r\ .$ (10a)

The radial tortoise coordinate $r^{*}$ is defined by $dr^{*}=\frac{dr}{f(r)}$.
A closed form expression of $r^{*}$ is

$\displaystyle
r^{*}=\frac{1}{2(r_{+}^{2}-r_{-}^{2})}\left\\{r_{+}\log\left(\frac{r-r_{+}}{r+r_{+}}\right)+r_{-}\log\left(\frac{r+r_{-}}{r-r_{-}}\right)\right\\}\
.$ (11)

The tortoise coordinate $r^{*}$ maps $[r_{+},\infty)$ to $(-\infty,0)$. Let us
also define another coordinate $r^{\\#}$ by
$\mathrm{d}r^{\\#}=\frac{n_{\phi}}{f}\mathrm{d}r$. $r^{\\#}$ also runs from
$-\infty$ to $0$ as one moves from horizon to infinity. Explicitly the
coordinate $r^{\\#}$ is

$\displaystyle r^{\\#}$
$\displaystyle=\frac{1}{2(r_{+}^{2}-r_{-}^{2})}\bigg{[}r_{-}\ln\bigg(\frac{r-r_{+}}{r+r_{+}}\bigg{missing})+r_{+}\ln\bigg(\frac{r+r_{-}}{r-r_{-}}\bigg{missing})\bigg{]}\
.$ (12)

In terms of these coordinates the ingoing Eddington-Finkelstein coordinates
become

$\displaystyle v=t+r^{*}\ ,\ \phi^{\prime}=\phi+r^{\\#}\ .$ (13a)

Let us now define a complexified tortoise coordinate $\chi$ (a rescaling of
$r^{*}$) by

$\displaystyle\chi(r)$
$\displaystyle=\oint_{r_{c}+i\varepsilon}^{r}\frac{dr^{\prime}}{\frac{i\beta}{2}f(r^{\prime})}$
$\displaystyle=\frac{1}{2\pi
i}\log\left\\{\left(\frac{r-r_{+}}{r+r_{+}}\right)\left(\frac{r+r_{-}}{r-r_{-}}\right)^{\frac{r_{-}}{r_{+}}}\left(\frac{r_{c}+r_{+}}{r_{c}-r_{+}}\right)\left(\frac{r_{c}-r_{-}}{r_{c}+r_{-}}\right)^{\frac{r_{-}}{r_{+}}}\right\\}\
.$ (14)

Note that the normalisation is chosen in such a way that
$\chi(r_{c}+i\varepsilon)=0$. $\chi(r)$ has branch points at $r_{+}$ and
$r_{-}$. We will consider branch cut in $\chi$ that extend from $r=r_{+}$ to
$r=\infty$. The inner horizon branch cut is chosen such that $\chi$ is
analytic in the real interval between the two horizons. Since we will be
interested in the region outside the future horizon $r_{+}$, we will now
consider the complexified doubled bulk spacetime as shown in Fig 3.

$r_{-}$$\times$$<$${\scriptsize{M_{L}}}$$>$$M_{R}$$r_{+}$$\qquad\qquad\qquad\qquad\qquad
r=r_{c}+i\varepsilon,\chi=0,\Theta=0$$\qquad\qquad\qquad\qquad\qquad
r=r_{c}-i\varepsilon,\chi=1,\Theta=\mu_{+}$ Figure 3: Picture of bulk at a
fixed $v$: red line is branch cut from horizon to boundary

We will glue two copies of rotating BTZ black holes across their future
horizon by a horizon cap and the monodromy in $\chi$ across the branch point
at $r=r_{+}$ will be one. Hence

$\chi(r_{c}+i\,\varepsilon)=0\,,\qquad\chi(r_{c}-i\,\varepsilon)=1\,.$ (15)

Let us also consider the following contour integral

$\displaystyle\Theta(r)$ $\displaystyle=\oint
n_{\phi}d\chi=\frac{2}{i\beta}\oint dr\frac{n_{\phi}}{f}$
$\displaystyle=\frac{1}{2\pi
i}\log\left\\{\left(\frac{r+r_{-}}{r-r_{-}}\right)\left(\frac{r-r_{+}}{r+r_{+}}\right)^{\frac{r_{-}}{r_{+}}}\left(\frac{r_{c}-r_{-}}{r_{c}+r_{-}}\right)\left(\frac{r_{c}+r_{+}}{r_{c}-r_{+}}\right)^{\frac{r_{-}}{r_{+}}}\right\\}\
.$ (16)

Note that $\Theta(r)$ goes from $0$ to $\mu_{+}$ in going from
$r_{c}+i\epsilon$ to $r_{c}-i\epsilon$ by picking up a monodromy across the
branch point at $r=r_{+}$, giving

$\Theta(r_{c}+i\,\varepsilon)=0\,,\qquad\Theta(r_{c}-i\,\varepsilon)=\mu_{+}\,.$
(17)

This explains the origin of the chemical potential $\mu_{+}$ arising from the
holographic contour in a rotating BTZ black hole. The metric rewritten in
ingoing $(v,\chi,\phi^{\prime})$ coordinates is given by

$\displaystyle ds^{2}$ $\displaystyle=-fdv^{2}+i\beta
fdvd\chi+r^{2}(d\phi^{\prime}-n_{\phi}dv)^{2}\ .$ (18)

The CFT contour for a given initial state at finite temperature and chemical
potential for angular momentum in this case is described as in Fig 4
(following Skenderis:2008dg ). In Skenderis:2008dg  the authors described how
the contour for the boundary CFT corresponding to a rotating black hole does
not only lie in the complex t plane, but also has a leg into the complex
$\phi$ plane. The solid circles in Fig 4 are to be identified. In this paper
we are interested in computing the generating functional for CFT correlators
holographically for a given initial state at finite temperature and chemical
potential due to angular momentum.

$\text{Im}(t)$$\text{Re}(t)$$\text{Im}(\phi)$ Figure 4: SK contour with an
initial state at finite temperature and chemical potential.

##  3 Probe scalar dynamics in bulk

###  3.1 Ingoing solution

In this section we consider a free massive probe scalar field $\Phi$ with mass
$M$ minimally coupled to the rotating BTZ black hole. According to AdS/CFT
correspondence, the scalar field in the bulk is dual to a scalar operator
$\mathcal{O}$ in the boundary CFT with conformal dimension
$\Delta=1+\sqrt{1+M^{2}}$.

The action for the probe scalar field is given by

$\displaystyle S$ $\displaystyle=-\oint d\chi
dvd\varphi^{\prime}\sqrt{-g}\left[\frac{1}{2}g^{AB}\partial_{A}\Phi\partial_{B}\Phi+\frac{M^{2}}{2}\Phi^{2}\right]\
.$ (19)

The classical scalar equation of motion is the Klein-Gordon equation

$\displaystyle\frac{1}{\sqrt{-g}}\partial_{A}[\sqrt{-g}g^{AB}\partial_{B}\Phi]-M^{2}\Phi=0\
.$ (20)

In the following we solve the massive scalar equation with appropriate
boundary conditions to find the ingoing bulk to boundary green’s function. To
solve the massive scalar equation, it is useful to define a new radial
coordinate $z$ as in Dias:2019ery

$\displaystyle z=\frac{r^{2}-r_{-}^{2}}{r_{+}^{2}-r_{-}^{2}}\ .$ (21)

In terms of the $z$ coordinate, the inner (Cauchy) horizon at $r=r_{-}$ is
located at $z=0$, the event horizon at $r=r_{+}$ is at $z=1$ and the
asymptotic boundary is at $z=+\infty$. We will consider a solution of the
scalar equation of the form

$\Phi_{in}=\int\frac{d\omega}{(2\pi)}\sum_{m}G^{+}(\omega,z,m)e^{i(m\phi^{\prime}-\omega
v)}\,.$ (22)

where $G^{+}(\omega,z,m)$ is the frequency domain retarded (ingoing) bulk to
boundary Green’s function. $\omega$ and $m$ are frequency and azimuthal number
of the mode. Written in z coordinate the equation of motion looks as following

$\displaystyle
z(z-1)\partial^{2}_{z}G^{+}+\bigg{[}2z-1-\frac{i\omega\\{z(r_{+}^{2}-r_{-}^{2})+r_{-}^{2}\\}^{1/2}}{r_{+}^{2}-r_{-}^{2}}+\frac{imr_{+}r_{-}}{(r_{+}^{2}-r_{-}^{2})\\{z(r_{+}^{2}-r_{-}^{2})+r_{-}^{2}\\}^{1/2}}\bigg{]}\partial_{z}G^{+}$
$\displaystyle-\frac{1}{4\\{z(r_{+}^{2}-r_{-}^{2})+r_{-}^{2}\\}^{1/2}}\bigg{[}i\omega+\frac{imr_{+}r_{-}}{\\{z(r_{+}^{2}-r_{-}^{2})+r_{-}^{2}\\}}+\frac{m^{2}}{\\{z(r_{+}^{2}-r_{-}^{2})+r_{-}^{2}\\}^{1/2}}\bigg{]}G^{+}-\frac{M^{2}}{4}G^{+}=0\,.$
(23)

If we consider an ansatz of the form

$\displaystyle G(z)$ $\displaystyle=e^{i(\omega
r^{*}-mr^{\\#})}z^{\alpha}(1-z)^{\beta}F(z)$ (24)

then the equation of motion in terms of F(z) becomes a hypergeometric
differential equation. We impose regularity at the horizon and normalisability
at the cut-off boundary to solve the equation of motion. The boundary
conditions are given by

$\displaystyle\frac{\mathrm{d}G^{+}(\omega,r,m)}{\mathrm{d}\chi}\bigg{\rvert}_{r=r_{+}}=0\
,\quad G^{+}\left(\omega,r_{c},m\right)=1\ .$ (25)

After using hypergeometric identities, the regular solution that satisfies the
above boundary conditions becomes (after writing the mass $M$ of the scalar in
terms of $\Delta$) (similar solution appears in a very recent paper
Natsuume:2020snz )

$\displaystyle G^{+}(z)=\frac{e^{i\left(\omega
r^{*}-mr^{\\#}\right)}z^{\frac{c-1}{2}-a+\frac{\Delta}{2}}(z-1)^{\frac{a+b-c}{2}}{}_{2}F_{1}\left(a-c+1-\frac{\Delta}{2},a-\frac{\Delta}{2},a+b-c+1;1-\frac{1}{z}\right)}{e^{i\left(\omega
r_{c}^{*}-mr_{c}^{\\#}\right)}z_{c}^{\frac{c-1}{2}-a+\frac{\Delta}{2}}(z_{c}-1)^{\frac{a+b-c}{2}}{}_{2}F_{1}\left(a-c+1-\frac{\Delta}{2},a-\frac{\Delta}{2},a+b-c+1;1-\frac{1}{z_{c}}\right)}$
(26)

where

$\displaystyle a$
$\displaystyle=\frac{1}{2}\bigg{(}2-\frac{i(\omega-m\mu_{-})}{\kappa_{-}}-\frac{i(\omega-m\mu_{+})}{\kappa_{+}}\bigg{)}$
(27a) $\displaystyle b$
$\displaystyle=\frac{1}{2}\bigg{(}\frac{-i(\omega-m\mu_{-})}{\kappa_{-}}-\frac{i(\omega-m\mu_{+})}{\kappa_{+}}\bigg{)}$
(27b) $\displaystyle c$
$\displaystyle=1-\frac{i(\omega-m\mu_{-})}{\kappa_{-}}\ .$ (27c)

This is the exact ingoing bulk to boundary green’s function that corresponds
to the infalling quasi normal modes of the black hole. The ingoing solution is
analytic in $z$ coordinate.

###  3.2 Solution on the SK contour

In this section, first we study the dynamics of the outgoing Hawking modes.
Within the holographic Schwinger-Keldysh path integral framework the outgoing
modes naturally arise as frequency and angular momentum reversed counter parts
of the ingoing modes. Hence the physics of the outgoing Hawking radiation is
captured by the advanced bulk to boundary green’s function. In the following
we will construct the advanced green’s function by imposing (generalised)
time-reversal transformation on the ingoing solution. Under (generalised) time
reversal transformation the coordinates and parameters transform as follows

$\displaystyle v$ $\displaystyle\rightarrow i\beta\chi-v$ (28a)
$\displaystyle\chi$ $\displaystyle\rightarrow\chi$ (28b)
$\displaystyle\phi^{\prime}$ $\displaystyle\rightarrow
i\beta\Theta-\phi^{\prime}$ (28c) $\displaystyle\omega$
$\displaystyle\rightarrow-\omega$ (28d) $\displaystyle m$
$\displaystyle\rightarrow-m\ .$ (28e)

These set of transformations keep the metric in equation (18) manifestly time
reversal invariant once the metric is written in the following way

$\displaystyle ds^{2}=-fdv\left(dv-i\beta
d\chi\right)-r^{2}\left(d\phi^{\prime}-n_{\phi}dv\right)\left(in_{\phi}\beta
d\chi-d\phi^{\prime}+n_{\phi}\left(dv-i\beta d\chi\right)\right)\ .$ (29)

Under time reversal, the ingoing solution $\Phi_{in}$ in equation (22) gets
mapped to the outgoing solution $\Phi_{out}$

$\displaystyle\Phi_{out}(v,z,\phi^{\prime})$
$\displaystyle=\int\frac{d\omega}{2\pi}\sum_{m}e^{-i\omega
v+im\phi^{\prime}}G(-\omega,z,-m)e^{-\beta\left(\omega\chi-m\Theta\right)}\ .$
(30a)

Note that the outgoing solution is non-analytic in $z$ coordinate because of
the presence of $\chi$ and $\Theta$ in the exponential factor. $\chi$ and
$\Theta$ have logarithmic branch point at $r=r_{+}$.

Now we construct the full solution on the doubled bulk spacetime by adding the
ingoing and outgoing solutions. The full solution captures the ingoing quasi
normal modes as well as the outgoing Hawking modes. The full solution on the
holographic SK contour is given by the following linear combination of the
ingoing and outgoing solutions

$\displaystyle\Phi$
$\displaystyle=\int\frac{d\omega}{2\pi}\frac{1}{2\pi}\sum_{m}\left\\{c_{\omega,m}\
G^{+}\left(\omega,z,m\right)+h_{\omega,m}\
G^{+}\left(-\omega,z,-m\right)e^{-\beta\left(\omega\chi-m\Theta\right)}\right\\}e^{-i\omega
v+im\phi^{\prime}}\ .$ (31)

At the boundary the doubled bulk spacetime asymptotes to CFT Schwinger-Keldysh
contour with left and right sources $\phi_{L}$ and $\phi_{R}$. Hence from the
requirement $\Phi\left(r_{c}+i\varepsilon\right)=\phi_{L}$ and
$\Phi\left(r_{c}-i\varepsilon\right)=\phi_{R}$, we find that

$\displaystyle c_{\omega,m}$ $\displaystyle=-f_{\omega,m}\
\phi_{L}\left(\omega,m\right)+\left(1+f_{\omega,m}\right)\phi_{R}\left(\omega,m\right)$
$\displaystyle h_{\omega,m}$
$\displaystyle=\left(1+f_{\omega,m}\right)\left(\phi_{L}\left(\omega,m\right)-\phi_{R}\left(\omega,m\right)\right)$
(32a)

where $f_{\omega,m}=\frac{1}{e^{\beta(\omega-m\mu_{+})}-1}$ is the Bose-
Einstein factor with chemical potential. Hence the full solution of the scalar
wave equation in equation (31) becomes

$\displaystyle\begin{split}\Phi&=\int\frac{d\omega}{2\pi}\frac{1}{2\pi}\sum_{m}\left\\{\left(-f_{\omega,m}\
\phi_{L}\left(\omega,m\right)+\left(1+f_{\omega,m}\right)\phi_{R}\left(\omega,m\right)\right)G^{+}\left(\omega,z,m\right)\right.\\\
&\left.+\left(1+f_{\omega,m}\right)\left(\phi_{L}\left(\omega,m\right)-\phi_{R}\left(\omega,m\right)\right)G^{+}\left(-\omega,z,-m\right)e^{-\beta\left(\omega\chi-m\Theta\right)}\right\\}e^{-i\omega
v+im\phi^{\prime}}\ .\end{split}$ (33)

We will use this exact solution to compute the quadratic influence functional
in section 4.1.

##  4 Influence functional of open scalar field

In this section we compute the quadratic and higher order influence
functionals of the open scalar field. To compute the influence functional it
is useful to work with the retarded-advance (RA) basis chaudhuri2019spectral ;
aurenche1992comparison ; van1994transformations  within the SK framework where
the KMS relations become manifest. The RA basis is defined as

$\displaystyle\phi_{F}\left(\omega,m\right)$
$\displaystyle=-\left(1+f_{\omega,m}\right)\phi_{R}\left(\omega,m\right)+f_{\omega,m}\phi_{L}\left(\omega,m\right)$
(34a) $\displaystyle\phi_{P}\left(\omega,m\right)$
$\displaystyle=-f_{\omega,m}\left(\phi_{R}\left(\omega,m\right)-\phi_{L}\left(\omega,m\right)\right)\
.$ (34b)

In terms of the RA basis we can write down the scalar solution given in
equation (33) as

$\displaystyle\Phi$
$\displaystyle=\int\frac{d\omega}{2\pi}\frac{1}{2\pi}\sum_{m}\left\\{-G^{+}\left(\omega,z,m\right)\phi_{F}+G^{+}\left(-\omega,z,-m\right)\phi_{P}e^{\beta\left(\omega\left(1-\chi\right)-m\left(\mu_{+}-\Theta\right)\right)}\right\\}e^{-i\omega
v+im\phi^{\prime}}\ .$ (35)

Using this scalar solution in RA basis, we compute the quadratic on-shell
action in section 4.1.

###  4.1 Quadratic effective theory

Imposing the Klein-Gordon equation of motion, the quadratic on-shell action of
the scalar field becomes

$\displaystyle S^{on-shell}$ $\displaystyle=-\oint d\chi
dvd\varphi^{\prime}\left\\{\partial_{A}\left(\frac{\sqrt{-g}}{2}g^{AB}\Phi\partial_{B}\Phi\right)\right\\}$
$\displaystyle=-\oint d\chi
dvd\phi^{\prime}\left\\{\partial_{\chi}\left(\frac{r}{2}\Phi\partial_{v}\Phi\right)+\partial_{\chi}\left(\frac{-ir}{\beta}\Phi\partial_{\chi}\Phi\right)+\partial_{\chi}\left(\frac{n_{\phi}r}{2}\Phi\partial_{\phi^{\prime}}\Phi\right)\right\\}$
$\displaystyle=-\int
dvd\phi^{\prime}\left[\frac{r}{2}\Phi\partial_{v}\Phi-\frac{ir}{\beta}\Phi\partial_{\chi}\Phi+\frac{n_{\phi}r}{2}\Phi\partial_{\phi^{\prime}}\Phi\right]\bigg{\rvert}^{\chi=1,\Theta=\mu_{+}}_{\chi=0,\Theta=0}\
.$ (36)

In the above equation we see that the quadratic on-shell action is a total
derivative, hence it has to be evaluated at the boundary. In RA basis the
quadratic influence functional becomes (see appendix A for details)

$\displaystyle S^{on-
shell}=\int\frac{d\omega}{2\pi}\frac{1}{2\pi}\sum_{m}\phi_{F}(\omega,m)\phi_{P}(-\omega,-m)\mathcal{G}_{FP}[\omega,m]$
(37)

where after setting $G^{+}(\omega,z_{c},m)=1$,

$\displaystyle\mathcal{G}_{FP}[\omega,m]$
$\displaystyle=-\frac{2ir_{c}}{\beta}\left[\partial_{\chi}G^{+}(\omega,z_{c},m)+\frac{\beta}{2}(\omega-
mn_{\phi})\right](1-e^{-\beta(\omega-\mu_{+}m)})\ .$ (38)

The FF and PP terms do not contribute to the radial integral since they are
purely analytic functions (the $\chi$ dependence goes away inside the
integral) and contributions from the left and right contours cancel each
other. We will compute the quadratic influence functional by substituting
$G^{+}$ in the above expression. Before proceeding with the calculation let us
define yet another basis known as the average-difference/Keldysh basis in
which the fluctuation-dissipation relation will be studied. This basis will be
more useful in the computation of higher order influence phase that will be
computed in a small frequency and momentum expansion. We define the average-
difference basis kamenev_2011 ; Keldysh:1964ud ; CHOU19851  in the following

$\displaystyle\phi_{F}\left(\omega,m\right)$
$\displaystyle=-\phi_{a}\left(\omega,m\right)-N_{\omega,m}\phi_{d}\left(\omega,m\right)$
(39a) $\displaystyle\phi_{P}\left(\omega,m\right)$
$\displaystyle=-\phi_{d}\left(\omega,m\right)f_{\omega,m}$ (39b)

where $N_{\omega,m}=f_{\omega,m}+\frac{1}{2}$ . In terms of the left and right
sources $\phi_{L}$ and $\phi_{R}$ at the boundary, the average source (mean
value) becomes $\phi_{a}=\frac{\phi_{R}+\phi_{L}}{2}$ and the difference
source (quantum/statistical fluctuations) becomes
$\phi_{d}=\phi_{R}-\phi_{L}$. The quadratic influence functional of equation
(37) written in average-difference basis becomes

$\displaystyle S^{on-shell}$
$\displaystyle=\int\frac{d\omega}{2\pi}\sum_{m}\left(\mathcal{G}_{ad}\phi_{a}\left(\omega,m\right)\phi_{d}\left(-\omega,-m\right)+\mathcal{G}_{dd}\phi_{d}\left(\omega,m\right)\phi_{d}\left(-\omega,-m\right)\right)$
(40)

where the coefficient functions that appear in front of the $\phi_{a}\phi_{d}$
and $\phi_{d}\phi_{d}$ terms are

$\displaystyle\begin{split}\mathcal{G}_{ad}&=2\frac{ir_{c}}{\beta}\left[\partial_{\chi}G^{+}(\omega,z_{c},m)+\frac{\beta}{2}(\omega-
mn_{\phi})\right]\ ,\end{split}$ (41)
$\displaystyle\begin{split}\mathcal{G}_{dd}&=\frac{ir_{c}}{2\beta}\frac{e^{\beta\left(\omega-m\mu_{+}\right)}+1}{e^{\beta\left(\omega-m\mu_{+}\right)}-1}\left[\partial_{\chi}G^{+}(\omega,z_{c},m)+\frac{\beta}{2}(\omega-
mn_{\phi})\right]+(\omega\rightarrow-\omega)\ .\end{split}$ (42)

where we have written $\mathcal{G}_{dd}$ in a symmetric way under
$\omega\rightarrow-\omega$. Note that there is no $\mathcal{G}_{aa}$ term in
the quadratic influence functional. This is a consequence of the microscopic
unitarity of the theory. After substituting for the solution of
$G^{+}\left(\omega,z_{c},m\right)$ from eq. (26) and picking up the
coefficient of $r_{c}^{4-2\Delta}$ we obtain the coefficient functions
$\mathcal{G}_{ad}$ and $\mathcal{G}_{dd}$. Note that this will be the result
after an appropriate counter term subtraction as is done in holographic
renormalisation. We finally obtain

$\displaystyle\mathcal{G}_{ad}[\omega,m]$
$\displaystyle=-\frac{2\left(r_{+}^{2}-r_{-}^{2}\right)^{\Delta-1}\Gamma^{2}\left(2-\Delta\right)\Gamma(\mathfrak{p}_{+})\Gamma(\mathfrak{p}_{-})\sin\left(\pi\Delta\right)}{\pi\Gamma\left(1-\Delta+\mathfrak{p}_{+}\right)\Gamma\left(1-\Delta+\mathfrak{p}_{-}\right)}$
(43a)
$\displaystyle=-\frac{2\left(r_{+}^{2}-r_{-}^{2}\right)^{\Delta-1}|\Gamma(\mathfrak{p}_{+})\Gamma(\mathfrak{p}_{-})|^{2}\sin\left(\pi(\Delta-\mathfrak{p}_{+})\right)\sin\left(\pi(\Delta-\mathfrak{p}_{-})\right)}{\pi\Gamma^{2}(\Delta-1)\sin(\pi\Delta)}$
(43b)
$\displaystyle=-\frac{2\left(r_{+}^{2}-r_{-}^{2}\right)^{\Delta-1}|\Gamma(\mathfrak{p}_{+})\Gamma(\mathfrak{p}_{-})|^{2}\left(\cosh\left(\frac{\beta_{-}\omega_{-}}{2}\right)-\cosh\left(\frac{\beta_{+}\omega_{+}}{2}-i\pi\Delta\right)\right)}{2\pi\Gamma^{2}(\Delta-1)\sin\left(\pi\Delta\right)}$
(43c)
$\displaystyle=-\frac{\left(r_{+}^{2}-r_{-}^{2}\right)^{\Delta-1}|\Gamma(\mathfrak{p}_{+})\Gamma(\mathfrak{p}_{-})|^{2}}{\pi\Gamma^{2}(\Delta-1)\sin\left(\pi\Delta\right)}$
(43d)
$\displaystyle\left(\cosh\left(\frac{\beta_{-}\omega_{-}}{2}\right)-\cosh\left(\frac{\beta_{+}\omega_{+}}{2}\right)\cos(\pi\Delta)+i\sinh\left(\frac{\beta_{+}\omega_{+}}{2}\right)\sin(\pi\Delta)\right)$
(43e)

where various parameters that appear in the above expression are defined as
follows

$\displaystyle\mathfrak{p}_{+}$
$\displaystyle=\frac{\Delta}{2}+i\frac{(m-\omega)}{2(r_{+}-r_{-})}$ (44a)
$\displaystyle\mathfrak{p}_{-}$
$\displaystyle=\frac{\Delta}{2}-i\frac{(m+\omega)}{2(r_{+}+r_{-})}$ (44b)
$\displaystyle\omega_{+}$
$\displaystyle=\omega-m\mu_{+}=\omega-m\frac{r_{-}}{r_{+}}$ (44c)
$\displaystyle\omega_{-}$
$\displaystyle=\omega-m\mu_{-}=\omega-m\frac{r_{+}}{r_{-}}$ (44d)
$\displaystyle\beta_{+}$ $\displaystyle=\frac{2\pi
r_{+}}{r_{+}^{2}-r_{-}^{2}}$ (44e) $\displaystyle\beta_{-}$
$\displaystyle=\frac{2\pi r_{-}}{r_{+}^{2}-r_{-}^{2}}\ .$ (44f)

$\mathfrak{p}_{+}$, $\mathfrak{p}_{-}$, $\omega_{+}$ and $\omega_{-}$ are
lightcone like dimensionless combination of frequency and momentum. From a
similar computation of $\mathcal{G}_{dd}$ we get

$\displaystyle\mathcal{G}_{dd}[\omega,m]$
$\displaystyle=\frac{-i}{2}\frac{\left(r_{+}^{2}-r_{-}^{2}\right)^{\Delta-1}|\Gamma(\mathfrak{p}_{+})\Gamma(\mathfrak{p}_{-})|^{2}}{\pi\Gamma^{2}(\Delta-1)}\cosh\left(\frac{\beta_{+}\omega_{+}}{2}\right)\
.$ (45)

By comparing equation (45) with equation (43e) we get the fluctuation-
dissipation relation in presence of chemical potential due to angular momentum
given by

$\displaystyle\mathcal{G}_{dd}=\frac{i}{2}\coth\left(\frac{\beta\left(\omega-m\mu_{+}\right)}{2}\right)Im(\mathcal{G}_{ad})\
.$ (46)

$Im(\mathcal{G}_{ad})$ gives the quadratic spectral function at finite
temperature CHOU19851 ; Haehl:2017eob .

Our quadratic influence functional results computed using a natural extension
of the recently proposed holographic SK prescription of Glorioso:2018mmw
match with the long known results of Son:2002sd  under following coordinate
transformations and redefinition of parameters

$\displaystyle x^{+}$ $\displaystyle=r_{+}t-r_{-}\phi\ ,\
x^{-}=-r_{-}t+r_{+}\phi\ ,$ (47a) $\displaystyle k_{+}$
$\displaystyle=\frac{\omega
r_{+}-mr_{-}}{r_{+}^{2}-r_{-}^{2}}=\frac{\beta_{+}\omega_{+}}{2\pi}$ (47b)
$\displaystyle k_{-}$ $\displaystyle=\frac{\omega
r_{-}-mr_{+}}{r_{+}^{2}-r_{-}^{2}}=\frac{\beta_{-}\omega_{-}}{2\pi}$ (47c)
$\displaystyle p_{+}$ $\displaystyle=\pi T_{L}\left(k_{+}+k_{-}\right)=\pi
T_{L}\frac{(\omega-m)}{(r_{+}-r_{-})}$ (47d) $\displaystyle p_{-}$
$\displaystyle=\pi T_{R}\left(k_{+}-k_{-}\right)=\pi
T_{R}\frac{(\omega+m)}{(r_{+}+r_{-})}$ (47e)

where $x^{\pm}$ are comoving coordinates, $k_{\pm}$ are momenta conjugate to
$x_{\pm}$ and $T_{L}$ and $T_{R}$ are the left and right temperatures given by

$\displaystyle T_{L}=\frac{r_{+}-r_{-}}{2\pi}\ ,\
T_{R}=\frac{r_{+}+r_{-}}{2\pi}\ .$ (48)

The answer given in Son:2002sd  for $\mathcal{G}_{ad}$ goes as

$\displaystyle\begin{split}\mathcal{G}_{ad}[\omega,m]&\sim\frac{|\Gamma\left(\frac{\Delta}{2}+i\frac{p_{+}}{2\pi
T_{L}}\right)\Gamma\left(\frac{\Delta}{2}+i\frac{p_{-}}{2\pi
T_{R}}\right)|^{2}}{\Gamma\left(2\frac{\Delta}{2}-1\right)^{2}\sin(2\pi\beta_{+})}\left[\cosh\left(\frac{p_{+}}{2T_{L}}-\frac{p_{-}}{2T_{R}}\right)\right.\\\
&\left.-\cos(2\pi\frac{\Delta}{2})\cosh\left(\frac{p_{+}}{2T_{L}}+\frac{p_{-}}{2T_{R}}\right)+i\sin\left(2\pi\frac{\Delta}{2}\right)\sinh\left(\frac{p_{+}}{2T_{L}}+\frac{p_{-}}{2T_{R}}\right)\right]\
.\end{split}$ (49)

###  4.2 Quartic order effective theory

In this section, we will turn on weak quartic self interaction of the massless
scalar field. At the leading order in the self interaction strength this term
corresponds to the four-point contact Witten diagram in the bulk. Going to
frequency domain we determine the quartic order influence functional of the
scalar field in a low frequency and angular momentum gradient expansion till
first sub leading order in derivatives. We are relying on derivative expanded
scalar solution to evaluate higher order influence phase rather than trying to
compute them using exact scalar solution since the later might be technically
somewhat complicated.

####  4.2.1 Quartic Influence functional

In the following we add a quartic self-interaction term to the original
action. We will compute the on-shell action by imposing the scalar equation of
motion and integrating out the complexified radial coordinate. We consider a
simpler setting where we have a massless self interacting scalar field in the
bulk 333The massless case is simpler since the scalar wave equation is simpler
and solving it order by order in perturbation parameter is easier. that is
dual to a marginal CFT operator with conformal dimension two at the boundary.
The action for the massless interacting scalar field is given by

$\displaystyle S$ $\displaystyle=-\int d\chi dv\int
d\phi^{\prime}\sqrt{-g}\left[\frac{1}{2}g^{AB}\partial_{A}\Phi\partial_{B}\Phi+\frac{\lambda}{4!}\Phi^{4}\right]$
(50)

where $\lambda$ is the interaction strength. The equation of motion of the
scalar field resulting from the above action is

$\displaystyle\frac{1}{\sqrt{-g}}\partial_{A}\left[\sqrt{-g}g^{AB}\partial_{B}\Phi\right]-\frac{\lambda}{3!}\Phi^{3}$
$\displaystyle=0\ .$ (51)

Since this equation is non-linear in $\Phi$, we will solve this equation
perturbatively in $\lambda$. We write the solution $\Phi$ as a series in
$\lambda$

$\displaystyle\Phi$ $\displaystyle=\sum_{n=0}^{\infty}\lambda^{n}\Phi_{n}\ .$
(52)

The action upto linear order in coupling constant $\lambda$ becomes

$\displaystyle S$ $\displaystyle=-\int dv\int d\phi^{\prime}\oint
d\chi\sqrt{-g}\left[\frac{g^{AB}}{2}\left(\partial_{A}\Phi_{0}\partial_{B}\Phi_{0}+2\lambda\partial_{A}\Phi_{1}\partial_{B}\Phi_{0}\right)+\frac{\lambda}{4!}\Phi_{0}^{4}\right]$
(53a) $\displaystyle\begin{split}&=-\int dv\int d\phi^{\prime}\oint
d\chi\sqrt{-g}\left[\frac{1}{2}\partial_{A}\left(\sqrt{-g}g^{AB}\left(\Phi_{0}+2\lambda\Phi_{1}\right)\partial_{B}\Phi_{0}\right)\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\left.-\frac{1}{2}\left(\Phi_{0}+2\lambda\Phi_{1}\right)\partial_{A}\left(\sqrt{-g}g^{AB}\partial_{B}\Phi_{0}\right)+\frac{\lambda}{4!}\Phi_{0}^{4}\right]\end{split}$
(53b) $\displaystyle S_{on-shell}$ $\displaystyle=-\int dv\int
d\phi^{\prime}\oint
d\chi\sqrt{-g}\left[\frac{1}{2}\partial_{A}\left(\sqrt{-g}g^{AB}\Phi_{0}\partial_{B}\Phi_{0}\right)+\frac{\lambda}{4!}\Phi_{0}^{4}\right]\
.$ (53c)

In the last line we have imposed the e.o.m of the free scalar $\Phi_{0}$ and
we choose $\Phi_{1}$ at the boundary to be zero as a choice of boundary
condition. The first term in the above equation is the quadratic on-shell
action that we have computed in the previous section. The quartic piece in the
on-shell action is

$\displaystyle S_{quartic,on-shell}$ $\displaystyle=-\int dv\int
d\phi^{\prime}\oint d\chi\sqrt{-g}\left[\frac{\lambda}{4!}\Phi_{0}^{4}\right]\
.$ (54)

We have to perform the radial integral in order to obtain the quartic
influence functional of the open scalar field at the boundary. To compute the
quartic integral, we will first solve the free scalar field $\Phi_{0}$ in a
gradient expansion in low frequency and angular momentum. We will then
substitute this solution in the action and integrate out the radial
coordinate. In the following we compute the derivative expansion of the free
scalar field.
Derivative expansion of scalar solution:
We will start by expanding the frequency domain bulk to boundary green’s
function in terms of $\beta\omega$ and $\beta m\mu$, both of them being
dimensionless parameters. We have denoted $\mu_{+}$ as $\mu$, this is the
notation we will follow now on. The derivative expansion is

$\displaystyle G^{+}$
$\displaystyle=G_{0}^{+}+\frac{\beta\omega}{2}G^{+}_{\omega}+\frac{\beta
m\mu}{2}G^{+}_{m}+\left(\frac{\beta\omega}{2}\right)^{2}G^{+}_{\omega^{2}}+\left(\frac{\beta
m\mu}{2}\right)^{2}G^{+}_{m^{2}}+\left(\frac{\beta\omega}{2}\right)\left(\frac{\beta
m\mu}{2}\right)G^{+}_{\omega m}+...$ (55)

where $G^{+}_{\omega}$, $G^{+}_{m}$ and all the higher order pieces are only
functions of the radial coordinate. In the following we determine the values
of $G_{0}^{+}$, $G_{\omega}^{+}$ and $G_{m}^{+}$ for the free massless scalar
field. Refer to appendix B for derivative expansion of massive scalar field.
We impose the following boundary conditions while writing the derivative
expansion of the green’s function. These boundary conditions correspond to
regularity of the green’s function at the horizon and its normalisability at
the UV cut-off $r_{c}$. The boundary conditions are

$\displaystyle\frac{dG_{0}^{+}}{d\chi}\rvert_{r_{+}}$ $\displaystyle=0\qquad
G^{+}_{0}\rvert_{r_{c}}=1$ (56a)
$\displaystyle\frac{d}{d\chi}G^{+}_{\omega^{n},m^{l}}$ $\displaystyle=0\qquad
G^{+}_{\omega^{n},m^{l}}\rvert_{r_{c}}=0\hskip 28.45274pt\forall n,l>0.$ (56b)

Eventually we will take $r_{c}\rightarrow\infty$ at the end of our
computations. At the leading order in derivative expansion i.e. at
$\mathcal{O}(\beta\omega)^{0}$ and $\mathcal{O}(\beta m\mu)^{0}$ the massless
scalar e.o.m gives

$\displaystyle\partial_{\chi}\left(r\partial_{\chi}G^{+}_{0}\right)$
$\displaystyle=0\ .$ (57)

After imposing regularity at the horizon and normalisability at the cut-off
boundary we get

$\displaystyle G^{+}_{0}=1\ .$ (58)

Collecting $\mathcal{O}(\frac{\beta\omega}{2})$ terms in the scalar e.o.m we
get

$\displaystyle\partial_{\chi}\left(r\partial_{\chi}G^{+}_{\omega}\right)+\partial_{\chi}r$
$\displaystyle=0\ .$ (59)

We integrate the above equation and impose the two boundary conditions to get

$\displaystyle\begin{split}G^{+}_{\omega}&=\frac{1}{i\pi}(1-\mu^{2})\left\\{-\frac{1}{2(\mu+1)}\log\left(\frac{r-r_{-}}{r_{c}-r_{-}}\right)+\frac{1}{2(\mu-1)}\log\left(\frac{r+r_{-}}{r_{c}+r_{-}}\right)\right.\\\
&\left.\qquad\qquad\qquad\
-\frac{1}{\mu^{2}-1}\log\left(\frac{r+r_{+}}{r_{c}+r_{+}}\right)\right\\}\
.\end{split}$ (60)

Since we are derivative expanding the ingoing solution that is analytic in the
radial variable, $G^{+}_{\omega}$ is also analytic between $r_{+}$ and
$r_{c}$. To find $G_{m}^{+}$, the required equation at
$\mathcal{O}(\frac{\beta m\mu}{2})$ that we need to solve is

$\displaystyle\mu\partial_{\chi}\left(r\partial_{\chi}G^{+}_{m}\right)-\partial_{\chi}\left(\frac{r_{+}r_{-}}{r}\right)=0\
.$ (61)

Solving the above equation and imposing the boundary conditions we obtain

$\displaystyle G_{m}^{+}$
$\displaystyle=\frac{1}{i\pi\mu}\left\\{-\frac{(1-\mu)}{2}\log\left(\frac{r-r_{-}}{r_{c}-r_{-}}\right)+\frac{(\mu+1)}{2}\log\left(\frac{r+r_{-}}{r_{c}+r_{-}}\right)-\mu\log\left(\frac{r+r_{+}}{r_{c}+r_{+}}\right)\right\\}\
.$ (62)

$G_{m}^{+}$ is analytic between $r_{+}$ and $r_{c}$ as well for the same
reason. Given the derivative expansion till linear order for the ingoing
green’s function, we obtain the full green’s function by adding the ingoing
and outgoing green’s functions where the outgoing part is just a frequency and
momentum reversed counter part of the derivative expanded ingoing solution.
For computational ease we will collect the even and odd parts of the full
Green’s function under frequency and momentum reversal and conveniently define
the following ‘even-odd’ basis where

$\displaystyle G_{even}=\frac{G^{+}(\omega,r,m)+G^{+}(-\omega,r,-m)}{2}\qquad
G_{odd}=\frac{G^{+}(\omega,r,m)-G^{+}(-\omega,r,-m)}{2}\ .$ (63)

In terms of this basis the full scalar solution can be written as

$\displaystyle\Phi$ $\displaystyle=G_{even}\phi_{even}+G_{odd}\phi_{odd}$ (64)

with

$\displaystyle\phi_{even}$
$\displaystyle=-\phi_{F}+e^{\beta\left(\omega\left(1-\chi\right)-m\mu\left(1-\tilde{\Theta}\right)\right)}\phi_{P}$
(65a)
$\displaystyle=\left[\phi_{a}-\left(\frac{e^{\beta\left(\omega(1-\chi)-m\mu(1-\tilde{\Theta})\right)}-1}{e^{\beta\left(\omega-m\mu\tilde{\Theta}\right)}-1}-\frac{1}{2}\right)\phi_{d}\right]$
(65b)
$\displaystyle=\left(1-e^{-\beta\left(\omega\chi-m\mu\tilde{\Theta}\right)}\right)\left(f_{\omega,m}+1\right)\phi_{R}+\left(e^{\beta\left(\omega(1-\chi)-m\mu(1-\tilde{\Theta})\right)}-1\right)f_{\omega,m}\phi_{L}\
,$ (65c) $\displaystyle\phi_{odd}$
$\displaystyle=-\phi_{F}-e^{\beta\left(\omega\left(1-\chi\right)-m\mu\left(1-\tilde{\Theta}\right)\right)}\phi_{P}$
(65d)
$\displaystyle=\left[\phi_{a}+\left(\frac{e^{\beta\left(\omega(1-\chi)-m\mu(1-\tilde{\Theta})\right)}+1}{e^{\beta\left(\omega-m\mu\tilde{\Theta}\right)}-1}+\frac{1}{2}\right)\phi_{d}\right]$
(65e)
$\displaystyle=\left(1+e^{-\beta\left(\omega\chi-m\mu\tilde{\Theta}\right)}\right)\left(f_{\omega,m}+1\right)\phi_{R}-\left(e^{\beta\left(\omega(1-\chi)-m\mu(1-\tilde{\Theta})\right)}+1\right)f_{\omega,m}\phi_{L}\
.$ (65f)

and we have normalised $\Theta$ in terms of $\tilde{\Theta}$ where
$\tilde{\Theta}=\frac{\Theta}{\mu}$. The full solution becomes non-analytic
due to the presence of $\chi$ and $\Theta$ in the outgoing solution. The full
solution upto second order in derivatives can be written as

$\displaystyle\Phi$
$\displaystyle=\left[1+\left(\frac{\beta\omega}{2}\right)^{2}G^{+}_{\omega^{2}}+\left(\frac{\beta
m\mu}{2}\right)^{2}G^{+}_{m^{2}}+\left(\frac{\beta\omega}{2}\right)\left(\frac{\beta
m\mu}{2}\right)G^{+}_{\omega m}\right]\phi_{even}$
$\displaystyle\quad+\left[G^{+}_{\omega}\left(\frac{\beta\omega}{2}\right)+G^{+}_{m}\frac{\beta
m\mu}{2}\right]\phi_{odd}+{\text{higher order terms}}\ .$ (66)

Now we obtain an important relation using equations (14), (16), (60) and (62)
that we explicitly use in the computation of quartic influence functional. The
relation is given by

$\displaystyle G_{\omega}^{+}+\chi$ $\displaystyle=-G_{m}^{+}+\tilde{\Theta}\
.$ (67)

Similar relation holds true even for the massive case. See appendix B for the
derivation of the relation in massive case.
Quartic Influence-functional:
Equipped with the derivative expanded full solution given in equation (66), we
now proceed to compute the quartic influence functional by substituting the
solution in equation (54). Substituting the derivative expansion for
$\phi_{even}$ the quartic influence functional in frequency domain becomes

$\displaystyle\begin{split}S_{quartic}&=\int\prod_{i=1}^{4}\frac{d\omega}{2\pi}\left(\frac{1}{2\pi}\right)^{4}\sum_{m_{1},m_{2},m_{3},m_{4}}\left\\{\mathcal{G}_{aaaa}\phi_{a}\left(\omega_{1},m_{1}\right)\phi_{a}\left(\omega_{2},m_{2}\right)\phi_{a}\left(\omega_{3},m_{3}\right)\phi_{a}\left(\omega_{4},m_{4}\right)\right.\\\
&\left.\qquad\qquad\qquad\qquad\qquad\mathcal{G}_{aaad}\
\phi_{a}\left(\omega_{1},m_{1}\right)\
\phi_{a}\left(\omega_{2},m_{2}\right)\phi_{a}\left(\omega_{3},m_{3}\right)\phi_{d}\left(\omega_{4},m_{4}\right)\right.\\\
&\left.\qquad\qquad\qquad\qquad\qquad+\mathcal{G}_{aadd}\
\phi_{a}\left(\omega_{1},m_{1}\right)\
\phi_{a}\left(\omega_{2},m_{2}\right)\phi_{d}\left(\omega_{3},m_{3}\right)\phi_{d}\left(\omega_{4},m_{4}\right)\right.\\\
&\left.\qquad\qquad\qquad\qquad\qquad\mathcal{G}_{addd}\
\phi_{a}\left(\omega_{1},m_{1}\right)\phi_{d}\left(\omega_{2},m_{2}\right)\phi_{d}\left(\omega_{3},m_{3}\right)\phi_{d}\left(\omega_{4},m_{4}\right)\right.\\\
&\left.\qquad\qquad\qquad\qquad\qquad+\mathcal{G}_{dddd}\
\phi_{d}\left(\omega_{1},m_{1}\right)\phi_{d}\left(\omega_{2},m_{2}\right)\phi_{d}\left(\omega_{3},m_{3}\right)\phi_{d}\left(\omega_{4},m_{4}\right)\right\\}\
.\end{split}$ (68)

The coefficient functions that enter in front of various terms in the quartic
on-shell action are given by the following radial integrals over complexified
radial coordinate. Note that we have performed the $v$ and $\phi^{\prime}$
integrals in the following expressions that result in total energy and
momentum conserving delta functions. The coefficient functions till first
order in derivative expansion are

$\displaystyle\mathcal{G}_{aaaa}$
$\displaystyle=\frac{\lambda}{4!}(2\pi)\delta\left(\sum_{i}\omega_{i}\right)(2\pi)\delta_{\sum_{i}m_{i}}\oint
d\chi\sqrt{-g}=0$ (69a) $\displaystyle\mathcal{G}_{aaad}$
$\displaystyle=\frac{\lambda}{3!}(2\pi)\delta\left(\sum_{i}\omega_{i}\right)(2\pi)\delta_{\sum_{i}m_{i}}\oint
d\chi\sqrt{-g}\left\\{X-\beta\frac{\left(\omega_{4}-\mu
m_{4}\right)}{2}\left(X^{2}-\frac{1}{4}\right)\right\\}$ (69b)
$\displaystyle\mathcal{G}_{aadd}$
$\displaystyle=\frac{\lambda}{4}(2\pi)\delta\left(\sum_{i}\omega_{i}\right)(2\pi)\delta_{\sum_{i}m_{i}}\oint
d\chi\sqrt{-g}\left\\{X^{2}-\beta\left(\omega_{4}-\mu
m_{4}\right)\left(X^{3}-\frac{X}{4}\right)\right\\}$ (69c)
$\displaystyle\begin{split}\mathcal{G}_{addd}&=\frac{\lambda}{3!}(2\pi)\delta\left(\sum_{i}\omega_{i}\right)(2\pi)\delta_{\sum_{i}m_{i}}\oint
d\chi\sqrt{-g}\left\\{X^{3}+\frac{\beta(\omega_{4}-\mu
m_{4})}{2}\left(X^{4}-\frac{X^{2}}{4}\right)\right\\}\end{split}$ (69d)
$\displaystyle\begin{split}\mathcal{G}_{dddd}&=\frac{\lambda}{4!}(2\pi)\delta\left(\sum_{i}\omega_{i}\right)(2\pi)\delta_{\sum_{i}m_{i}}\oint
d\chi\sqrt{-g}X^{4}\end{split}$ (69e)

where $X$ inside the integrals is independent of $\omega$ and $m$ and is
defined as

$\displaystyle X=G_{\omega}^{+}+\chi-\frac{1}{2}\ .$ (70)

We have used the relation (67) in obtaining the above integral expressions for
the coefficient functions. $\mathcal{G}_{aaaa}=0$ since the radial integral is
completely analytic. This is a consequence of microscopic unitarity or
equivalently a consequence of Schwinger-Keldysh collapse rule that states that
under collapse of two consecutive legs of the SK contour, the effective action
vanishes.
Evaluation of quartic integrals:
To evaluate the above integrals we note that the complexified contour integral
can be broken into two parts as given below

$\displaystyle\oint
d\chi=\int_{r_{c}}^{r_{+}}dr\frac{dr}{d\chi}+\int_{r_{+}}^{r_{c}}dr\frac{dr}{d\chi}$
(71)

where the integrand picks up a monodromy traversing across the branch point at
$r=r_{+}$. This monodromy is correctly incorporated by the following
substitution in the integrand as one goes across the branch point

$\displaystyle\chi-\frac{1}{2}$ $\displaystyle\rightarrow\chi+\frac{1}{2}$
(72a) $\displaystyle\tilde{\Theta}-\frac{1}{2}$
$\displaystyle\rightarrow\tilde{\Theta}+\frac{1}{2}\ .$ (72b)

Since there are no other poles of the integrand at $r=r_{+}$, the above
substitution in the integrand gives the value of the radial integration.

We evaluate the integrals in the slow rotation limit as a series in $\mu$. We
will only keep terms that are linear order in $\mu$. The ‘bare’ values of the
integrals till linear order in small rotation and linear order in derivative
expansion are the following

$\displaystyle\mathcal{G}_{aaaa}$ $\displaystyle=0$ (73a)
$\displaystyle\mathcal{G}_{aaad}$
$\displaystyle=\frac{\lambda}{3!}(2\pi)\delta\left(\sum_{i}\omega_{i}\right)(2\pi)\delta_{\sum_{i}m_{i}}\left\\{\frac{r_{c}^{2}}{2}-\frac{r_{+}^{2}}{2}+\frac{\beta}{2\pi}\left(\omega_{4}-\mu
m_{4}\right)\log\left(\frac{r_{c}}{r_{+}}\right)\right\\}+\mathcal{O}(\mu^{2})$
(73b) $\displaystyle\mathcal{G}_{aadd}$
$\displaystyle=\frac{\lambda}{4}(2\pi)\delta\left(\sum_{i}\omega_{i}\right)(2\pi)\delta_{\sum_{i}m_{i}}\left\\{\frac{i}{\pi}r_{+}^{2}\log\left(\frac{r_{c}}{r_{+}}\right)-\frac{\beta}{8}r_{+}^{2}\left(\omega_{4}-\mu
m_{4}\right)\right\\}+\mathcal{O}(\mu^{2})$ (73c)
$\displaystyle\begin{split}\mathcal{G}_{addd}&=\frac{\lambda}{3!}(2\pi)\delta\left(\sum_{i}\omega_{i}\right)(2\pi)\delta_{\sum_{i}m_{i}}\left\\{\frac{r_{c}^{2}}{8}-\frac{r_{+}^{2}}{4}+\frac{i\beta}{8\pi}\left(\omega_{4}-\mu
m_{4}\right)\left(\frac{6}{\pi^{2}}r_{+}^{2}\zeta[3]-r_{+}^{2}\log\left(\frac{r_{c}}{r_{+}}\right)\right)\right\\}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\
+\mathcal{O}(\mu^{2})\end{split}$ $\displaystyle\mathcal{G}_{dddd}$
$\displaystyle=\frac{\lambda}{4!}(2\pi)\delta\left(\sum_{i}\omega_{i}\right)(2\pi)\delta_{\sum_{i}m_{i}}\left\\{-ir_{+}^{2}\frac{3\zeta[3]}{2\pi^{3}}+i\frac{r_{+}^{2}}{2\pi}\log\left(\frac{r_{c}}{r_{+}}\right)\right\\}+\mathcal{O}(\mu^{2})\
.$ (73d)

The values of the integrals are divergent. Hence we need to introduce counter
terms to kill these divergences and get finite answers. We do a minimal
subtraction where we only kill the divergent pieces and keep the finite part
as it is. From the above values of quartic influence functional, one clearly
sees divergences that are both of polynomial type and logarithmic type in
$r_{c}$. The polynomial divergences can be removed by adding local counter
terms to the theory. However, the logarithmic divergences can not be removed
by local counter terms since the argument of the logarithm has dependence on
both $r_{c}$ and $r_{+}$. It turns out that, to renormalise the influence
phase in the massless case, one has to do an additional source renormalisation
Jana:2020vyx  that removes the logarithmic divergences. In the massive case
one only encounters polynomial divergences that can be removed by local
counter terms, hence source renormalisation is not needed.

In the following we discuss the divergence structure of the massless integrals
only till linear order in small $\mu$ and linear order in derivative
expansion. We will then find the counter terms (till linear order in small
$\mu$ and linear order in derivative expansion), that cancel the divergences
of the bare integrals. The details of the source renormalisation is discussed
in section 4.4 when we consider arbitrary higher order terms in the effective
action. Note that the divergence of the influence phase comes from integrals
of the form Jana:2020vyx

$\displaystyle F_{k}$ $\displaystyle=\oint
d\chi\sqrt{-g}\left(X+G_{\omega}-\frac{1}{2}\right)^{k}\ .$ (74)

The divergences of $F_{k}$ for $k=1,2,3,4$ till linear order in $\mu$ are of
the forms

$\displaystyle F_{1}$ $\displaystyle\sim\frac{r_{c}^{2}}{2}$ (75a)
$\displaystyle F_{2}$
$\displaystyle\sim\frac{ir_{+}^{2}}{\pi}\ln\left(\frac{r_{c}}{r_{+}}\right)$
(75b) $\displaystyle F_{3}$ $\displaystyle\sim\frac{r_{c}^{2}}{8}$ (75c)
$\displaystyle F_{4}$
$\displaystyle\sim\frac{ir_{+}^{2}}{2\pi}\ln\left(\frac{r_{c}}{r_{+}}\right)\
.$ (75d)

We add the following counter terms (till linear order in $\mu$) to regularise
the integrals

$\displaystyle\delta_{1}$ $\displaystyle=-\frac{r_{c}^{2}}{2}$ (76a)
$\displaystyle\delta_{2}$
$\displaystyle=-\frac{ir_{+}^{2}}{\pi}\ln\left(\frac{r_{c}}{r_{+}}\right)$
(76b) $\displaystyle\delta_{3}$ $\displaystyle=-\frac{r_{c}^{2}}{8}$ (76c)
$\displaystyle\delta_{4}$
$\displaystyle=-\frac{ir_{+}^{2}}{2\pi}\ln\left(\frac{r_{c}}{r_{+}}\right)\ .$
(76d)

Here $\delta_{1}$ and $\delta_{2}$ are local counter terms. However
$\delta_{2}$ and $\delta_{4}$ are not. The origin of $\delta_{2}$ and
$\delta_{4}$ lies within the source renormalisation scheme that we discuss in
section 4.4.

###  4.3 General $n$-th order effective theory

In this section we extend our study of effective theory to general n-point
self interaction of the scalar field. We will compute the $n$-th order
influence functional following the same recipe discussed in the previous
section. The coefficient functions appearing in front of various terms of the
effective action are expressed in terms of integrals over the complexified
radial contour. We will now consider the more complicated massive scalar case
for our analysis since we will express the coefficient functions in the
effective theory only as radial integrals. The action of the scalar field is
given by

$\displaystyle S$ $\displaystyle=-\int d\chi
dvd\phi\sqrt{-g}\left\\{\frac{1}{2}\partial_{A}\Phi
g^{AB}\partial_{B}\Phi+M^{2}\Phi^{2}+\frac{\lambda_{n}}{n!}\Phi^{n}\right\\}\
.$ (77)

The influence phase is again given by the on-shell action. Like quartic on-
shell action, the $n$-point on-shell action becomes

$\displaystyle S_{n-point}$ $\displaystyle=-\int d\chi
dvd\phi\sqrt{-g}\frac{\lambda_{n}}{n!}\Phi_{0}^{n}$ (78)

where $\Phi_{0}$ satisfies the homogeneous wave equation

$\displaystyle\frac{1}{\sqrt{-g}}\partial_{A}\left(\sqrt{-g}g^{AB}\partial_{B}\Phi_{0}\right)-M^{2}\Phi_{0}=0\
.$ (79)

The derivation of this equation is exactly same as given in the previous
section. We will write down the influence phase in position space upto first
order in derivatives. Let us start with an expression for the $\Phi_{0}$ in
position space written in average-difference/Keldysh basis

$\displaystyle\begin{split}\Phi_{0}&=G_{0}^{+}\left(1+\frac{i\beta}{2}\left(\tilde{G}_{\omega}^{+}\partial_{v}-\mu\tilde{G}_{m}^{+}\partial_{\phi^{\prime}}\right)\right)\left\\{\phi_{a}^{b}+\frac{i\beta}{8}\left(\partial_{v}-\mu\partial_{\phi^{\prime}}\right)\phi_{d}^{b}+\phi_{d}^{b}\left(\chi+\tilde{G}_{\omega}^{+}-\frac{1}{2}\right)\right.\\\
&\left.\hskip
142.26378pt-\frac{i\beta}{2}\left(\partial_{v}-\mu\partial_{\phi^{\prime}}\right)\phi_{d}^{b}\left(\chi+\tilde{G}_{\omega}^{+}-\frac{1}{2}\right)^{2}\right\\}+{\text{higher
order terms}}\ .\end{split}$ (80)

The superscript ‘b’ in the above expression denotes that the sources at
infinity are bare sources. We substitute equation (80) in equation (78) to
obtain

$\displaystyle\begin{split}\frac{\Phi_{0}^{n}}{n!}&=(G_{0}^{+})^{n}\sum_{k=0}^{n}\frac{\left(\phi_{a}^{b}+\frac{i\beta}{8}\left(\partial_{v}-\mu\partial_{\phi^{\prime}}\right)\phi_{d}^{b}\right)^{n-k}}{(n-k)!}\frac{\left(\phi_{d}^{b}X-\frac{i\beta}{2}\left(\partial_{v}-\mu\partial_{\phi^{\prime}}\right)\phi_{d}^{b}X^{2}\right)^{k}}{k!}\\\
&\quad+\partial_{t}\left\\{\frac{i\beta}{2(n!)}(G_{0}^{+})^{n}\tilde{G}_{\omega}^{+}\left(\phi_{a}^{b}+\phi_{d}^{b}X\right)^{n}\right\\}-\mu\partial_{\phi^{\prime}}\left\\{\frac{i\beta}{2(n!)}(G_{0}^{+})^{n}\tilde{G}_{m}^{+}\left(\phi_{a}^{b}+\phi_{d}^{b}X\right)\right\\}\end{split}$
(81)

where

$\displaystyle X=\tilde{G}_{\omega}^{+}+\chi-\frac{1}{2}$ (82)

and $\tilde{G}_{\omega}^{+}=\frac{G_{\omega}^{+}}{G_{0}^{+}}$ for a massive
scalar field. Since the last two terms in equation (81) are total derivatives,
they do not contribute in the influence phase. Expanding the terms inside the
sum upto first order in derivatives we get

$\displaystyle\begin{split}\frac{\Phi_{0}^{n}}{n!}&=(G_{0}^{+})^{n}\sum_{k=0}^{n}\frac{\left((\phi_{a}^{b})^{n-k}+(n-k)(\phi_{a}^{b})^{n-k-1}\frac{i\beta}{8}\left(\partial_{v}-\mu\partial_{\phi^{\prime}}\right)\phi_{d}^{b}\right)}{(n-k)!}\\\
&\qquad\qquad\qquad\frac{\left((\phi_{d}^{b})^{k}X^{k}-k(\phi_{d}^{b})^{k-1}X^{k+1}\frac{i\beta}{2}\left(\partial_{v}-\mu\partial_{\phi^{\prime}}\right)\phi_{d}^{b}\right)}{k!}\end{split}$
(83a)
$\displaystyle\begin{split}&=(G_{0}^{+})^{n}\left\\{\sum_{k=0}^{n}\frac{(\phi_{a}^{b})^{n-k}(\phi_{d}^{b})^{k}}{(n-k)!k!}X^{k}+\sum_{k=0}^{n-1}\frac{(\phi_{a}^{b})^{n-k-1}(\phi_{d}^{b})^{k}}{(n-k-1)!k!}X^{k}\frac{i\beta}{8}\left(\partial_{v}-\mu\partial_{\phi^{\prime}}\right)\phi_{d}^{b}\right.\\\
&\qquad\qquad\left.-\sum_{k=1}^{n}\frac{(\phi_{a}^{b})^{n-k}(\phi_{d}^{b})^{k-1}}{(n-k)!(k-1)!}X^{k+1}\frac{i\beta}{2}\left(\partial_{v}-\mu\partial_{\phi^{\prime}}\right)\phi_{d}^{b}\right\\}+{\text{higher
order}}\ .\end{split}$ (83b)

Hence the bare influence functional becomes 444Note that $v$ goes to $t$ at
the boundary where $r^{*}$ goes to $0$, also $\phi^{\prime}$ goes to $\phi$
since $r^{\\#}$ goes to $0$ at the boundary.

$\displaystyle\begin{split}S^{b}&=-\lambda_{n}\int
dtd\phi\left\\{\sum_{k=1}^{n}\frac{(\phi_{a}^{b})^{n-k}(\phi_{d}^{b})^{k}}{(n-k)!k!}\mathcal{F}^{b}_{n,k}+\sum_{k=1}^{n-1}\frac{(\phi_{a}^{b})^{n-k}(\phi_{d}^{b})^{k-1}}{(n-k)!(k-1)!}\mathcal{F}^{b}_{n,k-1}\frac{i\beta}{8}\left(\partial_{t}-\mu\partial_{\phi}\right)\phi_{d}^{b}\right.\\\
&\left.\hskip
56.9055pt-\sum_{k=1}^{n-1}\frac{(\phi_{a}^{b})^{n-k}(\phi_{d}^{b})^{k-1}}{(n-k)!(k-1)!}\mathcal{F}^{b}_{n,k+1}\frac{i\beta}{2}\left(\partial_{t}-\mu\partial_{\phi}\right)\phi_{d}^{b}\right\\}\end{split}$
(84)

where

$\displaystyle\mathcal{F}^{b}_{n,k}=\int_{r_{+}}^{r_{c}}dr\>r\>(G_{0}^{+})^{n}\left\\{\left(\chi+\tilde{G}_{\omega}^{+}+\frac{1}{2}\right)^{k}-\left(\chi+\tilde{G}_{\omega}^{+}-\frac{1}{2}\right)^{k}\right\\}\
.$ (85)

We have also used the fact that $k=0$ term in the second sum in equation (83b)
vanishes after integrating over the holographic Schwinger Keldysh contour
since the integrand is analytic. $k=n$ term in the third sum in this equation
also does not contribute since that term is just a total derivative. The bare
influence functional that is computed with the bare sources as boundary
conditions is divergent. The divergence structure can be understood for
massive case by looking at the asymptotics of the solution as discussed in
appendix B. We cancel the divergences employing an appropriate renormalisation
scheme as discussed in the next section.

###  4.4 Renormalizing the $n$-th order effective theory

The function $\mathcal{F}^{b}_{n,k}$ in equation (85) is divergent. The
divergence structures are discussed below. Massive case:
The divergence coming from $X^{2k+1}$ is

$\displaystyle\mathcal{F}^{b}_{n,2k+1}=\mathcal{F}^{r}_{n,2k+1}+\frac{\Lambda_{\Delta}}{4^{k}}\
,\quad\Lambda_{\Delta}=r_{+}^{2}\left(\frac{\tilde{\zeta}_{c}}{1-\mu^{2}}\right)^{(\nu-1)n}\frac{\tilde{\zeta}_{c}}{2(\nu-1)n+2}$
(86)

where $\tilde{\mathcal{\zeta}}=\left(\frac{r}{r_{+}}\right)^{2}$ and rescaled
conformal dimension $\nu=\frac{\Delta}{2}$ . Hence this is divergent for all
$\nu$.
For $X^{2k}$ i.e. for $\mathcal{F}^{b}_{n,2k}$ there is no divergence for
$\nu<1$ that are relevant deformations in the CFT (see equation (131) of
appendix B).
Massless case:
The divergence coming from $X^{2k+1}$ is

$\displaystyle\mathcal{F}^{b}_{n,2k+1}$
$\displaystyle=\mathcal{F}^{r}_{n,2k+1}+\frac{\delta}{4^{k}}$ (87)

and for $X^{2k}$ it is

$\displaystyle\mathcal{F}^{b}_{n,2k}$
$\displaystyle=\mathcal{F}^{r}_{n,2k}+k\frac{\eta}{4^{k-1}}$ (88)

where the divergences are

$\displaystyle\delta=\frac{r_{c}^{2}}{2}\
,\qquad\eta=\frac{i(r_{+}^{2}-r_{-}^{2})}{\pi}\ln\left(\frac{r_{c}}{r_{+}}\right)\
.$ (89)

For the massive case we add local counter terms in the bare influence phase to
cancel the polynomial divergences and obtain a finite influence phase. However
the massless case is more tricky, since along with polynomial divergences we
also get non local logarithmic divergences that can not be removed by adding
local state independent counter terms in the bare influence phase. It turns
out that if we renormalise the sources at the cutoff in a particular way such
that the bare and renormalised sources agree when the cutoff is taken to
infinity, the logarithmic divergences get cancelled and the renormalised
effective action becomes finite Jana:2020vyx . This source renormalisation
prescription for the massless case is given by the following temperature
dependent dressing of sources where the difference source mixes with the
average source

$\displaystyle\phi^{b}_{a}=\phi^{r}_{a}-\frac{\eta}{2\delta}\phi^{r}_{d}\
,\qquad\phi^{b}_{d}=\phi^{r}_{d}+\frac{\eta}{2\delta}i\beta\left(\partial_{t}-\mu\partial_{\phi}\right)\phi^{r}_{d}\
.$ (90)

Note that as we take $r_{c}\to\infty$,
$\lim_{r_{c}\to\infty}\frac{\eta}{\delta}=0$, i.e., the bare and the
renormalised sources match at asymptotic infinity. We consider the following
counter term expressed in terms of state independent renormalised sources

$\displaystyle S^{c.t}=\lambda_{n}\int
dtd\phi\frac{\delta}{n!}\left[\left(\phi^{r}_{a}+\frac{\phi_{d}^{r}}{2}\right)^{n}-\left(\phi^{r}_{a}-\frac{\phi_{d}^{r}}{2}\right)^{n}\right]\
.$ (91)

Then the statement is that when we add this to the bare influence phase the
renormalised influence functional is free of nonlocal logarithmic divergences.
The renormalised influence functional becomes

$\displaystyle S^{r}$
$\displaystyle=\lim_{r_{c}\rightarrow\infty}\left(S^{b}+S^{c.t}\right)$ (92)
$\displaystyle\begin{split}&=-\lambda_{n}\int
dtd\phi\left\\{\sum_{k=1}^{n}\frac{(\phi_{a}^{r})^{n-k}(\phi_{d}^{r})^{k}}{(n-k)!k!}\mathcal{F}^{r}_{n,k}+\sum_{k=1}^{n-1}\frac{(\phi_{a}^{r})^{n-k}(\phi_{d}^{r})^{k-1}}{(n-k)!(k-1)!}\mathcal{F}^{r}_{n,k-1}\frac{i\beta}{8}\left(\partial_{t}-\mu\partial_{\phi}\right)\phi_{d}^{r}\right.\\\
&\left.\hskip
56.9055pt-\sum_{k=1}^{n-1}\frac{(\phi_{a}^{r})^{n-k}(\phi_{d}^{r})^{k-1}}{(n-k)!(k-1)!}\mathcal{F}^{r}_{n,k+1}\frac{i\beta}{2}\left(\partial_{t}-\mu\partial_{\phi}\right)\phi_{d}^{r}\right\\}\
.\end{split}$ (93)

##  5 Nonlinear Langevin dynamics in rotating plasma

The effective theory of the open scalar field has a description in terms of a
nonlinear Langevin equation with non-Gaussianities in the thermal
fluctuations. In the following we discuss the dual stochastic description of
the effective dynamics.

###  5.1 Deriving effective action via stochastic path integral

We start with an ansatz for the stochastic dynamics governed by the non-linear
Langevin equation in presence of rotation for the average field $\psi_{a}$ as
follows (for non-rotating case see Jana:2020vyx )

$\displaystyle\mathcal{E}[\psi_{a},\eta]$ $\displaystyle\equiv f\eta$ (94)
$\displaystyle\begin{split}\mathcal{E}[\psi_{a},\eta]&\equiv-K\partial_{t}^{2}\psi_{a}+D\partial_{\phi}^{2}\psi_{a}+F\partial_{t}\partial_{\phi}\psi_{a}+\gamma\left(\partial_{t}-\mu\partial_{\phi}\right)\psi_{a}+\bar{\mu}\psi_{a}\\\
&+\sum_{k=1}^{n-1}\left(\theta_{k}\frac{\eta^{k-1}}{k!}\frac{\psi_{a}^{n-k}}{(n-k)!}+\bar{\theta}_{k}\frac{\eta^{k-1}}{k!}\frac{\psi_{a}^{n-k-1}}{(n-k-1)!}\left(\partial_{t}-\mu\partial_{\phi}\right)\psi_{a}\right)\
.\end{split}$ (95)

The stochastic variable $\eta$ is the thermal noise, $f$ is the strength of
the noise, $Z$, $X$ and $Y$ introduce colors in the thermal noise, $\gamma$ is
the damping constant, $\theta_{k}$ and $\bar{\theta}_{k}$ correspond to the
thermal jitter on top of the damping constant and anharmonicities in the
dynamics of the open scalar field. We consider the most general probability
distribution upto two derivatives acting on the stochastic noise $\eta$. We
take the distribution of the noise to be

$\displaystyle\mathcal{P}[\eta]$ $\displaystyle=\mathcal{N}e^{-\int
dtd\phi\left(\frac{f}{2!}\eta^{2}+\frac{Z}{2!}(\partial_{t}\eta)^{2}+\frac{X}{2!}(\partial_{\phi}\eta)^{2}+\frac{Y}{2!}\partial_{t}\eta\partial_{\phi}\eta+\frac{\theta_{n}}{n!}\eta^{n}\right)}$
(96)

where $\mathcal{N}$ is a normalisation factor and $\theta_{n}$ is the non-
Gaussianity in the noise. In the following we obtain an effective action from
a stochastic path integral that we eventually compare with the influence
functional obtained from holographic computations. This will determine the
parameters of the non-linear Langevin equation in terms of the effective
couplings computed from holography. To compute the effective action from
stochastic path integral we will exploit Martin-Siggia-Rose (MSR)
1973PhRvA…8..423M ; 1978PhRvB..18..353D ; 1976ZPhyB..23..377J  trick. For this
we will have to introduce an auxiliary field $\psi_{d}$ and convert following
identity

$\displaystyle 1$ $\displaystyle=\int D\psi_{a}D\eta\
\delta\left(\mathcal{E}[\psi_{a},\eta]-f\eta\right)\mathcal{P}[\eta]$ (97)

to the identity

$\displaystyle 1$ $\displaystyle=\int D\psi_{a}D\psi_{d}D\eta e^{-i\int
d^{2}x\left(\mathcal{E}[\psi_{a},\eta]-f\eta\right)\psi_{d}}\mathcal{P}[\eta]\
.$ (98)

Now we integrate over the noise $\eta$ to get an effective action in terms of
$\psi_{a}$ and $\psi_{d}$. This can be done by shifting
$\eta\rightarrow\eta+i\psi_{d}$ and taking $\eta\rightarrow 0$. This will give
the leading order Schwinger-Keldysh effective action with non-zero rotation

$\displaystyle\begin{split}S_{\psi}&=-\int
dtd\phi\left[-K\psi_{d}\partial_{t}^{2}\psi_{a}+D\psi_{d}\partial_{\phi}^{2}\psi_{a}+F\psi_{d}\partial_{t}\partial_{\phi}\psi_{a}+\bar{\mu}\psi_{a}\psi_{d}+\gamma\>\psi_{d}\left(\partial_{t}-\mu\partial_{\phi}\right)\psi_{a}\right.\\\
&\left.\hskip
85.35826pt-\frac{if}{2!}\psi_{d}^{2}+\frac{iZ}{2!}(\partial_{t}\psi_{d})^{2}+\frac{iX}{2!}(\partial_{\phi}\psi_{d})^{2}+\frac{iY}{2!}\partial_{t}\psi_{d}\partial_{\phi}\psi_{d}\right.\\\
&\left.\hskip
56.9055pt-i\sum_{k=1}^{n}\theta_{k}\frac{(i\psi_{d})^{k}}{k!}\frac{\psi_{a}^{n-k}}{(n-k)!}-i\sum_{k=1}^{n-1}\bar{\theta}_{k}\frac{(i\psi_{d})^{k}}{k!}\frac{\psi_{a}^{n-k-1}}{(n-k-1)!}\left(\partial_{t}-\mu\partial_{\phi}\right)\psi_{a}\right]\
.\end{split}$ (99)

###  5.2 Comparing with holographic result

In this section we will compare the effective action derived from stochastic
path integral to the effective action computed using holography. This way we
evaluate the parameters in the non-linear Langevin equation in terms of the
effective couplings computed from holography. The way to do this is by
uplifting the sources $\phi_{a}$ and $\phi_{d}$ that appear in the influence
functional to stochastic fields that we denote by $\psi_{a}$ and $\psi_{d}$.
Let us begin by writing down the exact quadratic influence functional

$\displaystyle\begin{split}S_{quadratic}&=\mathcal{N}_{2}\int
dtd\phi\left\\{i\frac{\psi_{d}^{2}}{\beta}+i\mathfrak{h}_{1,1}\partial_{t}\psi_{d}\partial_{\phi}\psi_{d}+i\mathfrak{h}_{0,2}\left(\partial_{\phi}\psi_{d}\right)^{2}+i\mathfrak{h}_{2,0}\left(\partial_{t}\psi_{d}\right)^{2}\right.\\\
&\left.+\mathfrak{g}_{0,0}\psi_{a}\psi_{d}-\left(\partial_{t}\psi_{a}-\mu\partial_{\phi}\psi_{a}\right)\psi_{d}+\mathfrak{g}_{0,2}\partial_{\phi}\psi_{a}\partial_{\phi}\psi_{d}+\mathfrak{g}_{2,0}\partial_{t}\psi_{a}\partial_{t}\psi_{d}+\mathfrak{g}_{1,1}\partial_{t}\psi_{a}\partial_{\phi}\psi_{d}\right\\}\end{split}$
(100)

where

$\displaystyle\mathcal{N}_{2}$
$\displaystyle=-\frac{(r_{+}^{2}-r_{-}^{2})^{\Delta-2}\Gamma\left(\frac{\Delta}{2}\right)^{4}r_{+}}{\Gamma(\Delta-1)^{2}}\
.$ (101)

The exact coefficient functions computed from holography are

$\displaystyle\mathfrak{h}_{0,2}$
$\displaystyle=\frac{\beta\left[\pi^{2}\mu^{2}-(1+\mu^{2})\mathcal{\psi}^{1}\left(\frac{\Delta}{2}\right)\right]}{8\pi^{2}}$
(102a) $\displaystyle\mathfrak{h}_{2,0}$
$\displaystyle=\frac{\beta\left[\pi^{2}-(1+\mu^{2})\mathcal{\psi}^{1}\left(\frac{\Delta}{2}\right)\right]}{8\pi^{2}}$
(102b) $\displaystyle\mathfrak{h}_{1,1}$
$\displaystyle=\frac{\beta\mu\left[\pi^{2}-2\mathcal{\psi}^{1}\left(\frac{\Delta}{2}\right)\right]}{4\pi^{2}}$
(102c) $\displaystyle\mathfrak{g}_{0,0}$
$\displaystyle=\frac{2\tan\left(\frac{\pi\Delta}{2}\right)}{\beta}$ (102d)
$\displaystyle\mathfrak{g}_{0,2}$
$\displaystyle=-\frac{\beta\left[\pi^{2}(-1+\mu^{2}\cos(\pi\Delta))\csc(\pi\Delta)+(1+\mu^{2})\mathcal{\psi}^{1}\left(\frac{\Delta}{2}\right)\tan\left(\frac{\pi\Delta}{2}\right)\right]}{4\pi^{2}}$
(102e) $\displaystyle\mathfrak{g}_{2,0}$
$\displaystyle=-\frac{\beta\left[\pi^{2}(-\mu^{2}+\cos(\pi\Delta))\csc(\pi\Delta)+(1+\mu^{2})\mathcal{\psi}^{1}\left(\frac{\Delta}{2}\right)\tan\left(\frac{\pi\Delta}{2}\right)\right]}{4\pi^{2}}$
(102f) $\displaystyle\mathfrak{g}_{1,1}$
$\displaystyle=\frac{\beta\mu\left[\pi^{2}-2\mathcal{\psi}^{1}\left(\frac{\Delta}{2}\right)\right]\tan\left(\frac{\pi\Delta}{2}\right)}{2\pi^{2}}$
(102g)

where $\mathcal{\psi}^{1}$ is the Polygamma function of order one.

The $n$-th order influence functional computed from holography is given by

$\displaystyle\begin{split}S^{r}_{\Phi^{n}}&=-\lambda_{n}\int
dtd\phi\left\\{\sum_{k=1}^{n}\frac{\left(\psi_{a}^{r}\right)^{n-k}\left(\psi_{d}^{r}\right)^{k}}{(n-k)!k!}\mathcal{F}_{n,k}^{r}\right.\\\
&\left.\qquad-\frac{i\beta}{2}\sum_{k=1}^{n-1}\frac{\left(\psi_{a}^{r}\right)^{n-k}\left(\psi_{d}^{r}\right)^{k-1}}{(n-k)!(k-1)!}\left(\partial_{t}-\mu\partial_{\phi}\right)\psi_{d}^{r}\>\left(\mathcal{F}_{n,k+1}^{r}-\frac{1}{4}\mathcal{F}_{n,k-1}^{r}\right)\right\\}\
.\end{split}$ (103)

Combining both the quadratic and $n$-th order influence functional we finally
have

$\displaystyle\begin{split}S&=\int
dtd\phi\left[\mathcal{N}_{2}\left\\{i\frac{\psi_{d}^{2}}{\beta}+i\mathfrak{h}_{1,1}\partial_{t}\psi_{d}\partial_{\phi}\psi_{d}+i\mathfrak{h}_{0,2}\left(\partial_{\phi}\psi_{d}\right)^{2}+i\mathfrak{h}_{2,0}\left(\partial_{t}\psi_{d}\right)^{2}+\mathfrak{g}_{0,0}\psi_{a}\psi_{d}\right.\right.\\\
&\hskip
28.45274pt\left.\left.-\left(\partial_{t}\psi_{a}-\mu\partial_{\phi}\psi_{a}\right)\psi_{d}+\mathfrak{g}_{0,2}\partial_{\phi}\psi_{a}\partial_{\phi}\psi_{d}+\mathfrak{g}_{2,0}\partial_{t}\psi_{a}\partial_{t}\psi_{d}+\mathfrak{g}_{1,1}\partial_{t}\psi_{a}\partial_{\phi}\psi_{d}\right\\}\right.\\\
&\left.-\lambda_{n}\sum_{k=1}^{n}\frac{\left(\psi_{a}^{r}\right)^{n-k}\left(\psi_{d}^{r}\right)^{k}}{(n-k)!k!}\mathcal{F}_{n,k}^{r}+\lambda_{n}\frac{i\beta}{2}\sum_{k=1}^{n-1}\frac{\left(\psi_{a}^{r}\right)^{n-k}\left(\psi_{d}^{r}\right)^{k-1}}{(n-k)!(k-1)!}\left(\partial_{t}-\mu\partial_{\phi}\right)\psi_{d}^{r}\>\left(\mathcal{F}_{n,k+1}^{r}-\frac{1}{4}\mathcal{F}_{n,k-1}^{r}\right)\right].\end{split}$
(104)

Now comparing equation (104) with equation (99) we obtain the parameters of
the non-linear Langevin equation in terms of the effective couplings computed
using holography. The values of the parameters of the linear Langevin equation
are

$\displaystyle K$ $\displaystyle=-\mathcal{N}_{2}\mathfrak{g}_{2,0}\qquad
D=\mathcal{N}_{2}\mathfrak{g}_{0,2}$ (105a) $\displaystyle F$
$\displaystyle=\mathcal{N}_{2}\mathfrak{g}_{1,1}\qquad\quad\bar{\mu}=-\mathcal{N}_{2}\mathfrak{g}_{0,0}$
(105b) $\displaystyle\gamma$ $\displaystyle=\mathcal{N}_{2}\qquad\qquad\quad
f=\frac{2\mathcal{N}_{2}}{\beta}$ (105c) $\displaystyle Z$
$\displaystyle=-2\mathcal{N}_{2}\mathfrak{h}_{2,0}\qquad
X=-2\mathcal{N}_{2}\mathfrak{h}_{0,2}$ (105d) $\displaystyle Y$
$\displaystyle=-2\mathcal{N}_{2}\mathfrak{h}_{1,1}\ .$ (105e)

Comparing the values of the parameters we see that the damping constant is
related to the fluctuation as below

$\displaystyle f$ $\displaystyle=\frac{2\gamma}{\beta}\ .$ (106)

This is the fluctuation dissipation relation between noise and dissipation in
the linear Langevin equation with rotation.

The extension to non-linear Langevin equation has additional parameters like
jitter in the damping, anharmonicity in the dynamics and non-Gaussianity in
the thermal noise. Those additional parameters are $\theta_{k}$ and
$\bar{\theta}_{k}$. From holographic computations we find

$\displaystyle\theta_{k}$
$\displaystyle=\frac{\lambda_{n}}{i^{k+1}}\mathcal{F}_{n,k}^{r}$ (107a)
$\displaystyle\bar{\theta}_{k}$
$\displaystyle=\frac{\beta\lambda_{n}}{2i^{k+2}}\left(\mathcal{F}_{n,k+1}^{r}-\frac{1}{4}\mathcal{F}_{n,k-1}^{r}\right)$
(107b)

where $\mathcal{F}_{n,k}^{r}$ are radial integrals that enter the effective
couplings. From these we get the generalised fluctuation dissipation relation
given by

$\displaystyle\frac{2}{\beta}\bar{\theta}_{k}$
$\displaystyle=\left(\theta_{k+1}+\frac{1}{4}\theta_{k-1}\right)\ .$ (108)

We have determined $\mathcal{F}_{n,k}^{r}$ for the quartic order effective
theory by performing the radial integrals analytically in the slow rotation
limit till linear order in rotation. In this case we find that generalised
fluctuation dissipation relation given in equation (108) holds true for linear
order in slow rotation.

##  6 Conclusion and Discussion

In this paper, we consider a natural extension of the gravitational Schwinger-
Keldysh path integral prescription of Glorioso:2018mmw  to rotating BTZ black
holes. The gravitational space-time asymptotes to the real-time Schwinger-
Keldysh contour of the dual rotating CFT at a given initial state with finite
temperature and chemical potential due to angular momentum. We study a probe
scalar field in the rotating BTZ geometry and obtain the ingoing quasi normal
modes and outgoing (time-reversed) Hawking radiation in section 3. By
computing the on-shell action and integrating over the complexified radial
coordinate we construct the effective theory of an open scalar field that is
coupled to a two-dimensional rotating thermal CFT plasma at the boundary in
section 4. The quadratic influence functionals computed in section 4.1 match
with the results of Son:2002sd  under appropriate coordinate transformations
and redefinition of parameters. The quadratic effective theory has a dual
stochastic description in terms of a linear Langevin equation in presence of
rotation. The noise and dissipation terms in the Langevin equation are related
by the fluctuation-dissipation relation with chemical potential.

We determine the higher order terms in the effective theory by derivative
expanding the free scalar solution in low frequency and angular momentum. The
coefficient functions that appear in front of the higher order terms are the
effective couplings evaluated from holography. We determine these coefficient
functions of higher order terms in the form of integrals over the bulk contour
in section 4.3. In the limit when the CFT plasma has low angular momentum i.e.
the dual BTZ black hole is slowly rotating, we compute the quartic order
integrals analytically till first order in slow rotation (similar thing can be
done for cubic effective theory also) in section 4.2. We find the correct
renormalisation scheme to remove the divergences appearing in the values of
the bare integrals in section 4.4. The higher order effective theory has a
description in terms of a non-linear Langevin dynamics with non-Gaussianity in
the thermal fluctuations as obtained in section 5. We write down the
generalisation of the non-linear fluctuation dissipation relation in presence
of chemical potential (see Jana:2020vyx  for the non-rotating case).

It will be useful to do the integrals that appear as the coefficient functions
in the effective theory numerically. Employing a systematic holographic
renormalisation scheme as given in Skenderis:2008dg , will provide a better
handle on the divergences coming from evaluating the integrals in our
analysis. As a followup to our work, it will be interesting to generalise the
prescription to higher-dimensional rotating black holes. Using derivative
expansion in low frequency and angular momenta one can obtain the effective
theory holographically. It will also be interesting in higher dimension to add
a Chern-Simons term in the bulk since such a term in the bulk will lead to an
anomaly on the even dimensional rotating boundary CFT. Chern-Simons term in
the presence of rotation is expected to give rise to an anomaly induced
transport.

It will also be quite interesting to extend the holographic path integral
contour to near-extremal black holes and make connection with the derivative
expansion scheme given in moitra2020near  for near-extremal black branes.

## Acknowledgements

We are grateful to R. Loganayagam for suggesting the problem to us and having
several useful discussions during the course of this project. We would like to
thank Soumyadeep Chaudhury and Amitabh Virmani for reading the draft and
sending their comments. We would like to thank Mukund Rangamani and Amitabh
Virmani for pointing out relevant references. We also thank Chandan Jana,
Rohan Poojary and Akhil Sivakumar for discussing with us.

##  Appendix A Quadratic Influence functionals in RA basis

In the following we compute the quadratic influence functional in retarded-
advanced basis.

$\displaystyle S^{on-shell}$ $\displaystyle=-\int
dvd\varphi^{\prime}\left[\frac{r}{2}\Phi\partial_{v}\Phi-\frac{ir}{\beta}\Phi\partial_{\chi}\Phi+\frac{n_{\phi}r}{2}\Phi\partial_{\varphi^{\prime}}\Phi\right]\bigg{\rvert}^{\chi=1,\Theta=\mu_{+}}_{\chi=0,\Theta=0}$
(109a) $\displaystyle\begin{split}&=-\int
dvd\varphi^{\prime}\left[\int\frac{d\omega_{1}d\omega_{2}}{(2\pi)^{2}}\frac{1}{(2\pi)^{2}}\sum_{m_{1},m_{2}}\left(-G^{+}\left(\omega_{1},r,m_{1}\right)\phi_{F}\left(\omega_{1},m_{1}\right)\right.\right.\\\
&\left.\left.+G^{+}\left(-\omega_{1},r,-m_{1}\right)\phi_{P}\left(\omega_{1},m_{1}\right)e^{\beta\left(\omega_{1}\left(1-\chi\right)-m_{1}\left(\mu_{+}-\Theta\right)\right)}\right)e^{-i(\omega_{1}+\omega_{2})v+i(m_{1}+m_{2})\varphi^{\prime}}\right.\\\
&\left.\left(-\phi_{F}\left(\omega_{2},m_{2}\right)\left(\frac{r}{2}(-i\omega_{2})G^{+}\left(\omega_{2},r,m_{2}\right)-\frac{ir}{\beta}\partial_{\chi}G^{+}\left(\omega_{2},r,m_{2}\right)+\frac{n_{\phi}r}{2}(im_{2})G^{+}\left(\omega_{2},r,m_{2}\right)\right)\right.\right.\\\
&\left.\left.+\phi_{P}\left(\omega_{2},m_{2}\right)e^{\beta\left(\omega_{2}\left(1-\chi\right)-m_{2}\left(\mu_{+}-\Theta\right)\right)}\left(\frac{r}{2}(i\omega_{2})G^{+}\left(-\omega_{2},r,-m_{2}\right)-\frac{ir}{\beta}\partial_{\chi}G^{+}\left(-\omega_{2},r,-m_{2}\right)\right.\right.\right.\\\
&\left.\left.\left.-\frac{n_{\phi}r}{2}(im_{2})G^{+}\left(-\omega_{2},r,-m_{2}\right)\right)\right)\right]\bigg{\rvert}_{\chi=0,\Theta=0}^{\chi=1,\Theta=\mu_{+}}\
.\end{split}$ (109b)

Integrating over $v$ and $\phi^{\prime}$ coordinates, we get energy and
momenta conserving delta functions in the following

$\displaystyle\begin{split}S^{on-
shell}&=-\left[\int\frac{d\omega_{1}d\omega_{2}}{(2\pi)^{2}}\frac{1}{(2\pi)^{2}}\sum_{m_{1},m_{2}}(2\pi)\delta(\omega_{1}+\omega_{2})(2\pi)\delta_{m_{1}+m_{2},0}\right.\\\
&\left.\left\\{-G^{+}\left(\omega_{1},r,m_{1}\right)\phi_{F}\left(\omega_{1},m_{1}\right)\phi_{P}\left(\omega_{2},m_{2}\right)e^{\beta\left(\omega_{2}\left(1-\chi\right)-m_{2}\left(\mu_{+}-\Theta\right)\right)}\right.\right.\\\
&\left.\left.\left(\frac{r}{2}(i\omega_{2})G^{+}\left(-\omega_{2},r,-m_{2}\right)-\frac{ir}{\beta}\partial_{\chi}G^{+}\left(-\omega_{2},r,-m_{2}\right)-\frac{n_{\phi}r}{2}(im_{2})G^{+}\left(-\omega_{2},r,-m_{2}\right)\right)\right.\right.\\\
&\left.\left.-G^{+}(-\omega_{1},r,-m_{1})\phi_{P}(\omega_{1},m_{1})e^{\beta(\omega_{1}(1-\chi)-m_{1}(\mu_{+}-\Theta))}\phi_{F}(\omega_{2},m_{2})\right.\right.\\\
&\left.\left.\left(\frac{r}{2}(-i\omega_{2})G^{+}\left(\omega_{2},r,m_{2}\right)-\frac{ir}{\beta}\partial_{\chi}G^{+}\left(\omega_{2},r,m_{2}\right)+\frac{n_{\phi}r}{2}(im_{2})G^{+}\left(\omega_{2},r,m_{2}\right)\right)\right\\}\right]\bigg{\rvert}^{\chi=1,\Theta=\mu_{+}}_{\chi=0,\Theta=0}\
.\end{split}$ (110)

Since $\omega_{1}$ ,$\omega_{2}$ and $m_{1}$,$m_{2}$ are dummy variables, we
will interchange them in the last four terms

$\displaystyle\begin{split}S^{on-
shell}&=-\int\frac{d\omega_{1}d\omega_{2}}{(2\pi)^{2}}\frac{1}{(2\pi)^{2}}\sum_{m_{1},m_{2}}(2\pi)\delta(\omega_{1}+\omega_{2})(2\pi)\delta_{m_{1}+m_{2},0}\\\
&\left\\{-G^{+}\left(\omega_{1},r,m_{1}\right)\phi_{F}\left(\omega_{1},m_{1}\right)\phi_{P}\left(\omega_{2},m_{2}\right)e^{\beta\left(\omega_{2}\left(1-\chi\right)-m_{2}\left(\mu_{+}-\Theta\right)\right)}\right.\\\
&\left.\left(\frac{r}{2}(i\omega_{2})G^{+}\left(-\omega_{2},r,-m_{2}\right)-\frac{ir}{\beta}\partial_{\chi}G^{+}\left(-\omega_{2},r,-m_{2}\right)-\frac{n_{\phi}r}{2}(im_{2})G^{+}\left(-\omega_{2},r,-m_{2}\right)\right)\right.\\\
&\left.-G^{+}(-\omega_{2},r,-m_{2})\phi_{P}(\omega_{2},m_{2})e^{\beta(\omega_{2}(1-\chi)-m_{2}(\mu_{+}-\Theta))}\phi_{F}(\omega_{1},m_{1})\right.\\\
&\left.\left(\frac{r}{2}(-i\omega_{1})G^{+}\left(\omega_{1},r,m_{1}\right)-\frac{ir}{\beta}\partial_{\chi}G^{+}\left(\omega_{1},r,m_{1}\right)+\frac{n_{\phi}r}{2}(im_{1})G^{+}\left(\omega_{1},r,m_{1}\right)\right)\right\\}\bigg{\rvert}_{\chi=0,\Theta=0}^{\chi=1,\Theta=\mu_{+}}\
.\end{split}$ (111)

Imposing the energy and momentum delta functions, gives the influence phase in
the R-A basis

$\displaystyle\begin{split}S^{on-
shell}&=-\int\frac{d\omega}{2\pi}\frac{1}{2\pi}\sum_{m}2\phi_{F}\left(\omega,m\right)\phi_{P}\left(-\omega,-m\right)e^{-\beta\left(\omega\left(1-\chi\right)-m\left(\mu_{+}-\Theta\right)\right)}\\\
&\qquad\frac{ir}{\beta}\left\\{G^{+}\left(\omega,r,m\right)\left[\partial_{\chi}G^{+}\left(\omega,r,m\right)+\frac{\beta}{2}(\omega-
n_{\phi}m)G^{+}(\omega,r,m)\right]\right\\}\bigg{\rvert}_{\chi=0,\Theta=0}^{\chi=1,\Theta=\mu_{+}}\end{split}$
(112a)
$\displaystyle=\int\frac{d\omega}{2\pi}\frac{1}{2\pi}\sum_{m}\phi_{F}(\omega,m)\phi_{P}(-\omega,-m)\mathcal{G}_{FP}[\omega,m]\
.$ (112b)

where

$\displaystyle\mathcal{G}_{FP}[\omega,m]$
$\displaystyle=-\frac{2ir}{\beta}G^{+}(\omega,r_{c},m)\left[\partial_{\chi}G^{+}(\omega,r_{c},m)+\frac{\beta}{2}(\omega-
mn_{\phi})G^{+}(\omega,r_{c},m)\right](1-e^{-\beta(\omega-\mu_{+}m)})$ (113)

and other terms in the integral are not contributing because they are
analytic.

##  Appendix B Derivative expansion of massive scalar solution

The massive scalar field equation is given by

$\displaystyle\begin{split}&\partial_{\chi}\left(r\partial_{\chi}G^{+}\right)+\left(\frac{\beta\omega}{2}\right)\left(r\partial_{\chi}G^{+}+\partial_{\chi}\left(rG^{+}\right)\right)+\left(\frac{\beta
m}{2}\right)^{2}\frac{f}{r}G^{+}\\\ &-\left(\frac{\beta
m}{2}\right)2n_{\phi}(r)r\partial_{\chi}G^{+}-\frac{\beta
m}{2}\partial_{\chi}\left(rn_{\phi}(r)\right)G^{+}+\frac{\beta^{2}}{4}frM^{2}G^{+}=0\
.\end{split}$ (114)

We substitute the derivative expansion given in equation (55) in the above
equation and solve the equation order by order in $\beta\omega$ and $\beta
m\mu$. Zeroth order solution is obtained by solving

$\displaystyle\partial_{r}\left(rf\partial_{r}G_{0}^{+}\right)+rM^{2}G_{0}^{+}=0\
.$ (115)

Let us define a new variable and a parameter as follows

$\displaystyle\tilde{\mathcal{\zeta}}=\left(\frac{r}{r_{+}}\right)^{2}\
,\qquad\nu=\frac{\Delta}{2}\ .$ (116)

In terms of this (115) can be rewritten as

$\displaystyle\partial_{\tilde{\mathcal{\zeta}}}\left((\tilde{\mathcal{\zeta}}-\mu^{2})(\tilde{\mathcal{\zeta}}-1)\partial_{\tilde{\mathcal{\zeta}}}G_{0}^{+}\right)-\nu(\nu-1)G_{0}^{+}=0\
.$ (117)

The general solution to this

$\displaystyle
c_{1}P_{-\nu}\left(\frac{2\tilde{\mathcal{\zeta}}-1-\mu^{2}}{1-\mu^{2}}\right)+c_{2}Q_{-\nu}\left(\frac{2\tilde{\mathcal{\zeta}}-1-\mu^{2}}{1-\mu^{2}}\right)\
.$ (118)

Now imposing the boundary condition $\frac{dG_{0}^{+}}{d\chi}=0$ at $r_{+}$ we
remove the second solution. Normalizing the solution to one at the boundary
fixes $c_{1}$. Finally we obtain

$\displaystyle
G_{0}^{+}=\frac{P_{-\nu}\left(\frac{2\tilde{\mathcal{\zeta}}-1-\mu^{2}}{1-\mu^{2}}\right)}{P_{-\nu}\left(\frac{2\tilde{\mathcal{\zeta}}_{c}-1-\mu^{2}}{1-\mu^{2}}\right)}$
(119)

where $\tilde{\zeta}_{c}$ denotes the value $\tilde{\mathcal{\zeta}}$ at the
UV cut-off.

Now we move to the first order calculation:
$\tilde{G}_{\omega}^{+}=\frac{G_{\omega}^{+}}{G_{0}^{+}}$ satisfies

$\displaystyle\partial_{\tilde{\mathcal{\zeta}}}\left((\tilde{\mathcal{\zeta}}-\mu^{2})(\tilde{\mathcal{\zeta}}-1)(G_{0}^{+})^{2}\partial_{\tilde{\mathcal{\zeta}}}\tilde{G}_{\omega}^{+}\right)$
$\displaystyle=\frac{i(1-\mu^{2})}{2\pi}\partial_{\tilde{\mathcal{\zeta}}}\left((G_{0}^{+})^{2}\tilde{\mathcal{\zeta}}^{\frac{1}{2}}\right)\
.$ (120)

$\tilde{G}_{m}^{+}=\frac{G_{m}^{+}}{G_{0}^{+}}$ satisfies the following
equation

$\displaystyle\mu\partial_{\tilde{\mathcal{\zeta}}}\left((\tilde{\mathcal{\zeta}}-\mu^{2})(\tilde{\mathcal{\zeta}}-1)(G_{0}^{+})^{2}\partial_{\tilde{\mathcal{\zeta}}}\tilde{G}_{m}^{+}\right)$
$\displaystyle=\frac{1-\mu^{2}}{2\pi
i}\tilde{\mathcal{\zeta}}\left((G_{0}^{+})^{2}\tilde{\mathcal{\zeta}}^{-\frac{1}{2}}\mu\right)\
.$ (121)

Note that the boundary condition satisfied by $\tilde{G}^{+}$ are

$\displaystyle\frac{d}{d\chi}\tilde{G}^{+}_{\omega^{n},m^{l}}$
$\displaystyle=0\qquad\tilde{G}^{+}_{\omega^{n},m^{l}}\rvert_{r_{c}}=0\hskip
28.45274pt\forall n,l>0.$ (122)

Equation (120) can be integrated to obtain $\tilde{G}_{\omega}^{+}$

$\displaystyle\tilde{G}_{\omega}^{+}$
$\displaystyle=\frac{i(1-\mu^{2})}{2\pi}\int_{\tilde{\mathcal{\zeta}}_{c}}^{\tilde{\mathcal{\zeta}}}\frac{d\tilde{\mathcal{\zeta}}^{\prime}}{(\tilde{\mathcal{\zeta}}^{\prime}-\mu^{2})(\tilde{\mathcal{\zeta}}^{\prime}-1)(G_{0}^{+}[\tilde{\mathcal{\zeta}}^{\prime}])^{2}}\int_{1}^{\tilde{\mathcal{\zeta}}^{\prime}}\partial_{\tilde{\mathcal{\zeta}}^{\prime\prime}}\left((G_{0}^{+}[\tilde{\mathcal{\zeta}}^{\prime\prime}])^{2}(\tilde{\mathcal{\zeta}}^{\prime\prime})^{\frac{1}{2}}\right)$
(123a)
$\displaystyle=\frac{i(1-\mu^{2})}{2\pi}\int_{\tilde{\mathcal{\zeta}}_{c}}^{\tilde{\mathcal{\zeta}}}d\tilde{\mathcal{\zeta}}^{\prime}\left\\{\frac{\tilde{\mathcal{\zeta}}^{\prime\frac{1}{2}}}{(\tilde{\mathcal{\zeta}}^{\prime}-\mu^{2})(\tilde{\mathcal{\zeta}}^{\prime}-1)}-\frac{(G_{0}^{+}[1])^{2}}{(\tilde{\mathcal{\zeta}}^{\prime}-\mu^{2})(\tilde{\mathcal{\zeta}}^{\prime}-1)(G_{0}^{+}[\tilde{\mathcal{\zeta}}^{\prime}])^{2}}\right\\}\
.$ (123b)

Equation (121) can be integrated to obtain $\tilde{G}_{m}^{+}$

$\displaystyle\tilde{G}_{m}^{+}$ $\displaystyle=\frac{(1-\mu^{2})}{2\pi
i}\int_{\tilde{\mathcal{\zeta}}_{c}}^{\tilde{\mathcal{\zeta}}}\frac{d\tilde{\mathcal{\zeta}}^{\prime}}{(\tilde{\mathcal{\zeta}}^{\prime}-\mu^{2})(\tilde{\mathcal{\zeta}}^{\prime}-1)(G_{0}^{+}[\tilde{\mathcal{\zeta}}^{\prime}])^{2}}\int_{1}^{\tilde{\mathcal{\zeta}}^{\prime}}\partial_{\tilde{\mathcal{\zeta}}^{\prime\prime}}\left((G_{0}^{+}[\tilde{\mathcal{\zeta}}^{\prime\prime}])^{2}(\tilde{\mathcal{\zeta}}^{\prime\prime})^{-\frac{1}{2}}\right)$
(124a) $\displaystyle=\frac{(1-\mu^{2})}{2\pi
i}\int_{\tilde{\mathcal{\zeta}}_{c}}^{\tilde{\mathcal{\zeta}}}d\tilde{\mathcal{\zeta}}^{\prime}\left\\{\frac{\tilde{\mathcal{\zeta}}^{\prime-\frac{1}{2}}}{(\tilde{\mathcal{\zeta}}^{\prime}-\mu^{2})(\tilde{\mathcal{\zeta}}^{\prime}-1)}-\frac{(G_{0}^{+}[1])^{2}}{(\tilde{\mathcal{\zeta}}^{\prime}-\mu^{2})(\tilde{\mathcal{\zeta}}^{\prime}-1)(G_{0}^{+}[\tilde{\mathcal{\zeta}}^{\prime}])^{2}}\right\\}\
.$ (124b)

In fact one can give a general recursive solution to
$\tilde{G}_{\omega^{l},m^{k}}$ in terms of lower order functions

$\displaystyle\begin{split}\tilde{G}_{\omega^{l},m^{k}}^{+}&=\int_{\tilde{\mathcal{\zeta}}_{c}}^{\tilde{\mathcal{\zeta}}}\frac{1}{(\tilde{\mathcal{\zeta}}-\mu^{2})(\tilde{\mathcal{\zeta}}-1)(G_{0}^{+})^{2}}\int_{1}^{\tilde{\mathcal{\zeta}}^{\prime}}d\tilde{\mathcal{\zeta}}^{\prime\prime}\left\\{\frac{1-\mu^{2}}{2\pi
i}\left(2\tilde{\mathcal{\zeta}}^{\prime\prime-\frac{1}{2}}\mu\partial_{\tilde{\mathcal{\zeta}}^{\prime\prime}}G^{+}_{\omega^{l},m^{k-1}}+\partial_{\tilde{\mathcal{\zeta}}^{\prime\prime}}(\tilde{\mathcal{\zeta}}^{\prime\prime-\frac{1}{2}}\mu)G^{+}_{\omega^{l},m^{k-1}}\right.\right.\\\
&\left.\left.\hskip
85.35826pt-2\tilde{\mathcal{\zeta}}^{\prime\prime\frac{1}{2}}\partial_{\tilde{\mathcal{\zeta}}^{\prime\prime}}G^{+}_{\omega^{l-1},m^{k}}-G^{+}_{\omega^{l-1},m^{k}}\right)+\frac{(1-\mu^{2})^{2}}{(2\pi)^{2}}\frac{G^{+}_{\omega^{l},m^{k-2}}}{\tilde{\mathcal{\zeta}}^{\prime\prime}}\right\\}G_{0}^{+}[\tilde{\mathcal{\zeta}}^{\prime\prime}]\
.\end{split}$ (125)

Here we assumed that $G^{+}_{\omega^{l},m^{k}}$ with $l<0$ or $k<0$ is zero.
With this one can check that the general formula reproduces the special cases.
We also note an important relation that can be used to compute the massive
higher order influence functionals

$\displaystyle\tilde{G}_{\omega}^{+}+\chi$
$\displaystyle=-\tilde{G}_{m}^{+}+\tilde{\Theta}\ .$ (126)

This relation can be derived by using the following relations

$\displaystyle\chi$
$\displaystyle=\frac{-i(1-\mu^{2})}{2\pi}\int_{\tilde{\mathcal{\zeta}}_{c}}^{\tilde{\mathcal{\zeta}}}d\tilde{\mathcal{\zeta}}^{\prime}\frac{\tilde{\mathcal{\zeta}}^{\prime\frac{1}{2}}}{(\tilde{\mathcal{\zeta}}^{\prime}-\mu^{2})(\tilde{\mathcal{\zeta}}^{\prime}-1)}$
(127a) $\displaystyle\bar{\Theta}$ $\displaystyle=\frac{(1-\mu^{2})}{2\pi
i}\int_{\tilde{\mathcal{\zeta}}_{c}}^{\tilde{\mathcal{\zeta}}}d\tilde{\mathcal{\zeta}}^{\prime}\frac{\tilde{\mathcal{\zeta}}^{\prime-\frac{1}{2}}}{(\tilde{\mathcal{\zeta}}^{\prime}-\mu^{2})(\tilde{\mathcal{\zeta}}^{\prime}-1)}$
(127b) $\displaystyle\tilde{G}_{\omega}^{+}$
$\displaystyle=\frac{i(1-\mu^{2})}{2\pi}\int_{\tilde{\mathcal{\zeta}}_{c}}^{\tilde{\mathcal{\zeta}}}d\tilde{\mathcal{\zeta}}^{\prime}\left\\{\frac{\tilde{\mathcal{\zeta}}^{\prime\frac{1}{2}}}{(\tilde{\mathcal{\zeta}}^{\prime}-\mu^{2})(\tilde{\mathcal{\zeta}}^{\prime}-1)}-\frac{(G_{0}^{+}[1])^{2}}{(\tilde{\mathcal{\zeta}}^{\prime}-\mu^{2})(\tilde{\mathcal{\zeta}}^{\prime}-1)(G_{0}^{+}[\tilde{\mathcal{\zeta}}^{\prime}])^{2}}\right\\}$
(127c) $\displaystyle\tilde{G}_{m}^{+}$ $\displaystyle=\frac{(1-\mu^{2})}{2\pi
i}\int_{\tilde{\mathcal{\zeta}}_{c}}^{\tilde{\mathcal{\zeta}}}d\tilde{\mathcal{\zeta}}^{\prime}\left\\{\frac{\tilde{\mathcal{\zeta}}^{\prime-\frac{1}{2}}}{(\tilde{\mathcal{\zeta}}^{\prime}-\mu^{2})(\tilde{\mathcal{\zeta}}^{\prime}-1)}-\frac{(G_{0}^{+}[1])^{2}}{(\tilde{\mathcal{\zeta}}^{\prime}-\mu^{2})(\tilde{\mathcal{\zeta}}^{\prime}-1)(G_{0}^{+}[\tilde{\mathcal{\zeta}}^{\prime}])^{2}}\right\\}\
.$ (127d)

Let us define $x=\frac{2\tilde{\mathcal{\zeta}}-1-\mu^{2}}{1-\mu^{2}}$. Then
one can write the sum in the LHS or RHS of (126) as follows

$\displaystyle\tilde{G}_{\omega}^{+}+\chi$
$\displaystyle=\frac{1}{i\pi}\int_{x_{c}}^{x}\frac{1}{(x^{2}-1)P_{-\nu}(x)^{2}}$
$\displaystyle=\frac{i}{\pi}\frac{Q_{-\nu}(x)}{P_{-\nu}(x)}+i\cot(\pi\nu)\ .$
(128)

Using the derivative expanded solution for the massive case, let us analyse
the radial integrals as given in (85). For this, first let us note the
asymptotic behaviour of $G_{0}^{+}$ at large $\tilde{\zeta}$

$\displaystyle
P_{-\nu}\left(\frac{2\tilde{\zeta}-1-\mu^{2}}{1-\mu^{2}}\right)$
$\displaystyle\xrightarrow[\tilde{\zeta}\rightarrow\infty]{}\frac{\tilde{\zeta}^{\nu-1}}{1-\mu^{2}}\
.$ (129)

Using (129), we obtain the following asymptotic expression for the radial
integrals. For odd $k=2l+1$ from equation (85), we get

$\displaystyle\mathcal{F}^{b}_{n,2l+1}\sim\frac{1}{4^{k}}\int^{r_{c}}dr\>r\>(G_{0}^{+})^{n}\sim\frac{1}{4^{k}}r_{+}^{2}\left(\frac{\tilde{\zeta}_{c}}{1-\mu^{2}}\right)^{(\nu-1)n}\frac{\tilde{\zeta}_{c}}{2(\nu-1)n+2}\
.$ (130)

Hence this is divergent for $\nu>1-\frac{1}{n}$. For even $k=2l$, using (129)
and (128), we get the following divergence

$\displaystyle\mathcal{F}^{b}_{n,2k}\sim\frac{2k}{4^{k-1}}\int^{r_{c}}dr\>r\>(G_{0}^{+})^{n}\left(\chi+\tilde{G}_{\omega}^{+}\right)\sim\frac{2k}{4^{k-1}}\frac{1}{(1-\mu^{2})^{(\nu-1)n-2\nu+1}}\frac{r_{+}^{2}}{i\pi}\frac{\tilde{\zeta}_{c}^{(\nu-1)(n-2)}}{2(\nu-1)(n-2)(1-2\nu)}.$
(131)

This integral is thus divergent when $\nu>1$ for $n\geq 3$. However, for
$\nu<1$ that corresponds to relevant operator in the dual CFT, there is no
divergence for even $k$.

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