††institutetext: International Centre for Theoretical Sciences (ICTS-TIFR)
Tata Institute of Fundamental Research, Shivakote, Hesaraghatta Hobli,
Bengaluru 560089, India.

# Open EFT for Interacting Fermions from Holography

Godwin Martin, Shivam K. Sharma<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We initiate the study of an open EFT for finite-temperature holographic
systems with interacting fermions. In particular, we do this for Yukawa
interactions in the bulk using the real-time formalism (grSK geometry). From
the bulk perspective, this study corresponds to Yukawa scattering against a
black hole, incorporating the effects of Hawking radiation. We derive an EFT
in the exterior of the black hole and thus develop a Witten diagrammatic
understanding of the scattering processes. This allows the explicit evaluation
of boundary Schwinger-Keldysh (SK) correlators to arbitrary order, at tree-
level in the bulk. Here we present explicitly the SK generating functional up
to four-point functions. Finally, we represent the correlators in a column-
vector representation that manifests microscopic unitarity and thermality,
with all the statistical factors neatly folded in.

###### Keywords:

holography, open EFT, gravitational Schwinger-Keldysh, Yukawa interaction,
column-vector representation

##  1 Introduction

Understanding phases of matter without a quasiparticle description is a
challenging problem in physics varma2002singular ; Lee:2017njh , spanning from
the high-$T_{c}$ superconductors in condensed matter to the quark-gluon plasma
(QGP) in high-energy physics. The AdS/CFT correspondence Maldacena:1997re ;
Gubser:1998bc ; Witten:1998qj  provides a lifeline with solvable toy models
Hartnoll:2016apf ; Blake:2022uyo , offering a perturbative description through
gravitational duals. Thermal states in these holographic systems correspond to
black holes in gravity, simplifying complex questions in strongly coupled
systems.

Recent developments employ effective field theory (EFT) techniques
Polchinski:1992ed ; Polchinski:1993ii  in such holographic models Lee:2008xf ;
Lee:2009epi ; Liu:2009dm ; Faulkner:2009wj ; Iqbal:2011ae , focusing on the
dynamics of probes (e.g. electrons). This requires adopting the perspective of
open effective field theories (open EFTs). The aim here is to extract the
influence of the holographic system on the probe, captured by the Influence
functional Feynman:1963fq . This functional is commonly derived using real-
time/Schwinger-Keldysh formalism which we will review in the note.

Our goal in this note is to study an open EFT for the interacting fermions via
real-time holography Son:2002sd ; Skenderis:2008dg ; Skenderis:2008dh ;
Herzog:2002pc ; Son:2009vu . Real-time holography, in this context, refers to
a method similar to the Gibbons-Hawking prescription Gibbons:1976ue  for
computing real-time observables in gravitational systems. More precisely,
there exists a conjectured geometry Glorioso:2018mmw  that dominates the real-
time/Schwinger-Keldysh (SK) gravitational path integral. This geometry, known
as grSK, involves connecting two copies of the black hole exterior along their
future horizons. Advances in real-time holography have facilitated the
development of open EFTs, albeit limited to free probe fermions Giecold:2009tt
; Loganayagam:2020eue ; Loganayagam:2020iol . However, the inclusion of
interactions is essential as probe fermions are not free in reality.

This study examines the scattering of fermions against a black hole
background, considering the often overlooked influence of Hawking radiation
Hawking:1975vcx . The grSK geometry accurately accounts for the thermal
fluctuations of a black hole, i.e., Hawking radiation, and its validity across
various scenarios has been demonstrated by numerous studies
Chakrabarty:2019aeu ; Jana:2020vyx ; Loganayagam:2020eue ; Loganayagam:2020iol
; Colin-Ellerin:2020mva ; Ghosh:2020lel ; Bu:2020jfo ; He:2021jna ; He:2022jnc
; He:2022deg ; Loganayagam:2022zmq ; Pantelidou:2022ftm ; Loganayagam:2022teq
; Loganayagam:2023pfb . The framework extends successfully to the cases with
chemical potentials, such as charged Loganayagam:2020iol  and rotating black
holes Chakrabarty:2020ohe . Moreover, it inherently embodies the microscopic
unitarity and the thermality of holographic systems. In GodwinLoga , it has
been shown that grSK results for boundary correlators align with Witten
diagrammatics, suggesting an _exterior effective field theory_ for scalar
theories without any derivative interactions. It is worth noting that grSK
inherently provides an exterior EFT, a presumption made in earlier studies
Liu:2009dm ; Faulkner:2009wj ; Iqbal:2011ae . We aim to expand this
investigation by including fermionic fields.

To simplify the analysis, we work with the grSK geometry having a planar
horizon. More precisely, the AdSd+1 Schwarzschild blackbrane grSK geometry.
Furthermore, we assume that both the scalar and the spinor fields are
massless, interacting via Yukawa term. The dynamics of these fields in the
grSK geometry can be obtained perturbatively (in the Yukawa coupling). This
perturbative solution can then be used to compute the perturbative on-shell
action, which is the leading contribution111Here, we are referring to the
leading contribution in the large $N$ expansion of the boundary theory. to the
boundary SK generating functional.

Finally, it is important to highlight a key aspect of this note regarding our
approach to presenting results. Specifically, we express the SK generating
functional (or real-time correlators) in the column vector representation
Henning:1993gh ; Carrington:2006xj ; Chaudhuri:2018ymp . The column vector
representation closely resembles the Käll’en-Lehmann spectral representation
extended to thermal scenarios. This representation not only reveals the
intrinsic unitarity and thermality (Kubo-Martin-Schwinger conditions
Kubo:1957mj ; Martin:1959jp ) of the theory but also naturally includes all
the statistical factors (Bose-Einstein and Fermi-Dirac). While the scalar
column vector representation has been explored previously, there has not been
a generalization to spinors. In this note, we demonstrate that the results for
grSK, including spinors (specifically Yukawa interactions), naturally align
with a column vector representation. We conclude this introduction by
providing a brief outline of the subsequent sections.

### Outline

A brief outline of this article is as follows. In Sec. (2), we begin with a
review of the influence functional and its connection to the Schwinger-Keldysh
formalism. Here, we also briefly explain the gravitational counterpart of this
formalism. In Sec. (3), we consider Yukawa theory in the gravitational
Schwinger-Keldysh geometry and solve for its dynamics in a perturbative
expansion. In Sec. (4), we present an explicit demonstrative calculation,
calculating the three-point functions, and further present the Feynman rules
that come from grSK. Using these Feynman rules, we list all the Witten
diagrams that are relevant at the three point and the four-point functions.
Furthermore, we present the expressions for the generating functional
explicitly in the column vector representation in Sec. (5). We conclude in
Sec. (6) with a summary and a discussion of future directions. The appendices
provide some technical details and explicit expressions omitted from the main
text.

##  2 Influence Functional and grSK geometry

As discussed in the introduction, we aim to construct an open EFT for a system
of interacting bosonic and fermionic degree of freedom coupled to a
holographic thermal bath. A way of obtaining such an EFT was first given by
Feynman and Vernon Feynman:1963fq , sometimes dubbed as the Influence
functional formalism. The basic idea is to double the degrees of freedom to
describe the non-unitary evolution of system, which arises by integrating out
the bath.

The set-up we have is the following: consider a single system degree of
freedom $\mathfrak{X}$ coupled with many bath degrees of freedom $X_{i}$. The
total action of combined unitary theory is given as

$S_{\text{tot}}[\mathfrak{X},X_{i}]=\
S_{\text{s}}[\mathfrak{X}]+S_{\text{b}}[X_{i}]+S_{\text{s-b}}[\mathfrak{X},X_{i}]\
.$ (1)

where $S_{\text{s}}$, $S_{\text{b}}$ and $S_{\text{s-b}}$ are parts of the
action describing the system, the bath and the interaction between them
respectively. Integrating out the bath degrees of freedom $X_{i}$, we obtain
the non-unitary theory for the system. In the influence functional language,
the path integral becomes

$\int\mathcal{D}\mathfrak{X}\,\mathcal{D}X_{i}\,\exp\bigg{(}iS_{\text{tot}}[\mathfrak{X},X_{i}]\bigg{)}\mapsto\int\mathcal{D}\mathfrak{X}_{{}_{\text{R}}}\,\mathcal{D}\mathfrak{X}_{{}_{\text{L}}}\exp\bigg{(}iS_{\text{s}}[\mathfrak{X}_{{}_{\text{R}}}]-iS_{\text{s}}[\mathfrak{X}_{{}_{\text{L}}}]+iS_{\text{IF}}[\mathfrak{X}_{{}_{\text{R}}},\mathfrak{X}_{{}_{\text{L}}}]\bigg{)}\
,$ (2)

where $\mathfrak{X}_{{}_{\text{R}}}$ and $\mathfrak{X}_{{}_{\text{L}}}$ are
two copies of system degrees of freedom and
$S_{\text{IF}}[\mathfrak{X}_{{}_{\text{R}}},\mathfrak{X}_{{}_{\text{L}}}]$ is
the influence phase, containing the influence of the bath on the system.

The influence phase generates the real-time correlators of the operator
$\mathcal{O}$ coupled to the system degree of freedom $\mathfrak{X}$. More
precisely,
$S_{\text{IF}}[\mathfrak{X}_{{}_{\text{R}}},\mathfrak{X}_{{}_{\text{L}}}]$ is
the same as the real-time or the Schwinger-Keldysh generating functional of
the bath. The Schwinger-Keldysh (SK) formalism is the appropriate real-time
framework for addressing non-equilibrium questions in QFT kamenev_2011 ;
Bellac:2011kqa ; Rammer:2007zz . This is especially true if we want to
systematically account for fluctuations.

In SK formalism, we have a path-integral that evolves mixed states: this is
done through a closed time contour with a forward and backward time evolution.
Such a contour allows us to directly compute real-time correlators. In Fig.
(1), we depict the SK contour for a thermal state in the complex time plane.

$A$$B$$\mathcal{C}$(R)(L)$t_{i}$$t_{f}$$t_{i}-i\beta$$t_{f}-i\epsilon$ Figure
1: The SK contour at finite temperature $T=\frac{1}{\beta}$. The direction of
the arrow represents the direction of contour time $t_{\mathcal{C}}$,
involving both forward (R) and backward (L) time-evolving parts. Note that
points $A$ and $B$ are identified in the contour.

To be more concrete, consider a holographic conformal field theory in $d$
spacetime dimensions (CFTd), described by the action $S_{\text{CFT}}$. Let
$\mathcal{O}$ be a bosonic operator in this CFT, and $J$ be its source, both
living on the SK contour $\mathcal{C}$. The SK generating functional
$\mathcal{Z}_{\text{SK}}$ of the CFT is then given by the path-integral

$\mathcal{Z}_{\text{SK}}\left[J\right]=\int\mathcal{D}\mathcal{O}\
\exp\bigg{(}iS_{\text{CFT}}+i\oint\mathop{}\\!\mathrm{d}t_{{}_{\mathcal{C}}}\mathop{}\\!\mathrm{d}^{d-1}\textbf{x}\
J(x)\mathcal{O}(x)\bigg{)}\ .$ (3)

Assuming there are no sources on the imaginary part of the contour,222We have
not put any sources on the imaginary part of the contour as it would change
the initial thermal state. it is convenient to write fields on the right and
the left part separately. To do so, we introduce the following notation

$\displaystyle\mathcal{O}(t)\equiv\ \mathcal{O}_{{}_{\text{R}}}(t)\ ,$
$\displaystyle\quad J(t)\equiv\ J_{{}_{\text{R}}}(t)$
$\displaystyle\forall\quad t\in\mathbb{R}\ .$ (4)
$\displaystyle\mathcal{O}(t-i\epsilon)\equiv\ \mathcal{O}_{{}_{\text{L}}}(t)\
,$ $\displaystyle\quad J(t-i\epsilon)\equiv\ J_{{}_{\text{L}}}(t)$
$\displaystyle\forall\quad t\in\mathbb{R}\ .$

where subscript denotes the part of the contour on which fields live. We can
then write

$\mathcal{Z}_{\text{SK}}[J_{{}_{\text{R}}},J_{{}_{\text{L}}}]=\Bigg{\langle}\exp\Bigg{\\{}i\int\mathop{}\\!\mathrm{d}^{d}x\
\Big{[}J_{{}_{\text{R}}}(x)\mathcal{O}_{{}_{\text{R}}}(x)-J_{{}_{\text{L}}}(x)\mathcal{O}_{{}_{\text{L}}}(x)\Big{]}\
\Bigg{\\}}\Bigg{\rangle}_{\rm CFT}\ .$ (5)

By taking the functional derivatives of $\mathcal{Z}_{\text{SK}}$, we obtain
two-point Schwinger-Keldysh correlators $K_{{}_{AB}}$ as follows,

$iK_{{}_{AB}}(t_{1},t_{2})=\
\frac{1}{i^{2}}\frac{\delta^{2}\,\log\mathcal{Z}_{\text{SK}}}{\delta
J_{{}_{A}}(t_{1})\delta J_{{}_{B}}(t_{2})}\Bigg{|}_{J=0}\ .$ (6)

In the operator formalism, these SK correlators correspond to the contour-
ordered correlation functions. In contour ordering, time increases along the
arrow on the upper part of the contour and decreases along the arrow on the
lower part. Moreover, any point on the lower part is considered to have
greater contour time than any point on the upper part. Then, it is not hard to
check that the following holds,

$\begin{split}&iK_{{}_{RR}}(t_{1},t_{2})=\
\braket{\mathbb{T}\\{\mathcal{O}(t_{1})\mathcal{O}(t_{2})\\}}\ ,\qquad
iK_{{}_{RL}}(t_{1},t_{2})=\ \braket{\mathcal{O}(t_{2})\mathcal{O}(t_{1})}\
,\\\ &iK_{{}_{LL}}(t_{1},t_{2})=\
\braket{\overline{\mathbb{T}}\\{\mathcal{O}(t_{1})\mathcal{O}(t_{2})\\}}\
,\qquad iK_{{}_{LR}}(t_{1},t_{2})=\
\braket{\mathcal{O}(t_{1})\mathcal{O}(t_{2})}\ .\end{split}$ (7)

where $\mathbb{T}$ and $\overline{\mathbb{T}}$ represent the time ordering and
the anti-time ordering in real-time, respectively.

The SK formalism is robust as it applies from free to strongly interacting
systems. Fundamental principles like unitarity, causality and thermality are
manifest in it. The SK 2-point correlators (in the average-difference basis)
naturally encode the retarded, advanced and fluctuation responses of the field
theory. Here, we have only discussed the bosonic SK formulation but the whole
formalism can be easily generalized to fermions. Without going into the
details of all of this,333For a detailed discussion on SK formalism, we
recommend the readers to look to textbooks on non-equilibrium QFT kamenev_2011
; Bellac:2011kqa ; Rammer:2007zz . we now turn our attention to the
gravitational counterpart of the SK formalism.

In the gravitational dual, the thermal states of the Conformal Field Theory
(CFT) correspond to AdS black holes. The duplication of CFT fields directly
corresponds to the doubling of the AdS black hole geometry Skenderis:2008dg ;
Skenderis:2008dh . Then, the gravitational Schwinger-Keldysh (grSK) geometry
can be constructed from two copies of AdS black hole geometry such that it
gives SK contour in the boundary limit Glorioso:2018mmw . Hence, the SK
generating functional $\mathcal{Z}_{\text{SK}}$ can be computed by evaluating
the on-shell gravitational action Chakrabarty:2019aeu ; Jana:2020vyx .

The grSK geometry is obtained by glueing two copies of the blackbrane exterior
across their future horizons. Consider a blackbrane in asymptotically AdSd+1
background in ingoing Eddington-Finkelstein coordinates given by

$\mathop{}\\!\mathrm{d}s^{2}=\
-r^{2}f\,\mathop{}\\!\mathrm{d}v^{2}+2\,\mathop{}\\!\mathrm{d}v\,\mathop{}\\!\mathrm{d}r+r^{2}\mathop{}\\!\mathrm{d}\textbf{x}^{2}\
,\qquad\qquad f(r)=1-\left(\frac{r_{h}}{r}\right)^{d}.$ (8)

where $r_{h}$ is the horizon size. We first complexify the radial coordinate
resulting in a $(d+2)$-real dimensional manifold. The grSK geometry is then
defined as a codimension-1 slice of this manifold: the radial coordinate in
grSK varies along a contour in the complex radius plane as shown in Fig. (2)

$\blacktriangle$$\blacktriangleright$$\blacktriangleleft$$\blacktriangleright$$\blacktriangledown$(L)(R)$\mathbf{Re}\,\zeta=0$$\mathbf{Re}\,\zeta=1$$r_{h}$$\infty+i\epsilon\
(\zeta=0)$$\infty-i\epsilon\ (\zeta=1)$$\mathbf{Im}\,r$$\mathbf{Re}\,r$ Figure
2:  grSK geometry : The radial contour drawn on the complex $r$ plane at fixed
$v$, parametrized by coordinate $\zeta$.

We will find it convenient to parametrize this radial contour by a
complexified tortoise co-ordinate $\zeta$ defined via

$\frac{\mathop{}\\!\mathrm{d}\zeta}{\mathop{}\\!\mathrm{d}r}=\frac{2}{i\beta\,r^{2}f(r)}\
,$ (9)

where $\beta\equiv\frac{4\pi}{dr_{h}}$ is the inverse Hawking temperature of
the black brane. The coordinate $\zeta$ has a branch cut along the exterior
and is normalised to have a unit monodromy around the horizon branch point
$r=r_{h}$.

To gain a clearer insight into the grSK geometry, it is helpful to use Penrose
diagrams. These diagrams provide a clear visualization of how the grSK
geometry is constructed from the standard black brane geometry. The grSK
geometry is constructed in two steps, illustrated in Fig. (3).

1\. Take two AdS Schwarzschild blackbrane in ingoing EF coordinates.2\. Stitch
them across their future horizons, i.e. $\text{H}^{+}$.H+H+H-H-H-H+ Figure 3:
Schematic representation of the construction of the grSK geometry.

Here, we first consider the two exteriors of the AdS blackbrane geometry in
ingoing EF coordinates. Next, we connect these geometries across their future
horizons, resulting in the formation of the grSK geometry.

Regarding the diagram in Fig. (3), it is important to note a couple of
aspects. Firstly, the diagram does not explicitly depict what happens at the
future horizons during the stitching process. Additionally, it lacks a
connection to the cigar geometry, which represents the initial thermal state
in the CFT. Therefore, it’s crucial to view this Penrose diagram of the grSK
geometry as purely representational.444For a more in-depth understanding of
the Penrose diagram of the grSK geometry, readers are encouraged to look at
Jana:2020vyx . Concluding our discussion of the grSK geometry, it is worth
mentioning that this framework has been extended to include charged and
rotating black holes, as explored in Loganayagam:2020iol ; Chakrabarty:2020ohe
.

Having clarified the background geometry, the next question is: How do
different kinds of fields propagate and interact in this background? This
question has already been explored for a variety of holographic systems
Chakrabarty:2019aeu ; Jana:2020vyx ; Loganayagam:2020eue ; Loganayagam:2020iol
; Colin-Ellerin:2020mva ; Ghosh:2020lel ; Bu:2020jfo ; He:2021jna ; He:2022jnc
; He:2022deg ; Loganayagam:2022zmq ; Pantelidou:2022ftm ; Loganayagam:2022teq
; Loganayagam:2023pfb ; Chakrabarty:2020ohe ; GodwinLoga . However, for the
first time, we aim to investigate systems involving interacting fermions. As
an illustrative example in this context, we will consider both bosons and
fermions interacting through the Yukawa interaction.

##  3 Yukawa theory in grSK

We will now consider one of the simplest theories involving Dirac spinors,
i.e., Yukawa theory. For simplicity, here we will be concerned with a massless
Dirac spinor interacting with a massless scalar through the Yukawa
interaction. Furthermore, both of these will be minimally coupled to the grSK
spacetime, which itself will be non-dynamical.

Yukawa interactions in the bulk are described by the action

$S=S_{\rm free}+S_{\rm int}\ ,$ (10)

where $S_{\rm free}$ is the sum of the free massless Dirac action and the free
massless scalar action, and $S_{\rm int}$ is the Yukawa interaction term
defined as

$S_{\rm
int}=-i\oint\mathop{}\\!\mathrm{d}\zeta\int\mathop{}\\!\mathrm{d}^{d}x\sqrt{-g}\
\lambda\phi\bar{\Psi}\Psi\ .$ (11)

Here $\lambda$ is the Yukawa coupling constant that controls the strength of
the coupling between the Dirac spinor $\Psi$ and the scalar $\phi$. The symbol
$g$ denotes the determinant of the grSK metric. The explicit expression of the
free part of the above action will be described in detail below.

Before proceeding to understand this full action and its consequences, we will
first focus on the free part. We will describe scalar and spinor propagation
in the grSK geometry without any interaction terms between them. This will
also allow us to set up the necessary notation for the later sections. Later
in this section, as well as in the next section, we will use the notation thus
set up to describe the Yukawa interaction in detail.

###  3.1 Free scalar theory

We start with scalar propagation in the grSK geometry, i.e., we consider the
scalar part of the free action given above. We will see how to solve the
Klein-Gordon equation that comes from varying this action. The important
lesson here is that once we have one solution to the Klein-Gordon equation, we
then can use the time-reversal involution Loganayagam:2020eue  of the grSK
geometry to generate another independent solution. We will see this in more
detail soon.

The free scalar field is described by the action

$S_{\phi\rm free}=\
-\oint\mathop{}\\!\mathrm{d}\zeta\int\mathrm{d}^{d}x\sqrt{-g}\
\frac{1}{2}g^{MN}\partial_{M}\phi\partial_{N}\phi\ ,$ (12)

where $g^{MN}$ is the metric describing the grSK spacetime, and $g$ is the
determinant of this metric. Varying the above action, we obtain the equation
of motion

$\frac{1}{\sqrt{-g}}\partial_{M}\left(\sqrt{-g}g^{MN}\partial_{N}\right)\phi=0\
.$ (13)

The above partial differential equation can be reduced to a second-order
ordinary differential equation in the radial coordinate by passing to the
Fourier domain in the boundary coordinates. In other words, if we decompose
the field $\phi$ as

$\phi(\zeta,x)\equiv\int_{k}e^{ikx}\phi(\zeta,k)\
,\qquad\int_{k}\equiv\int\frac{\mathop{}\\!\mathrm{d}^{d}k}{(2\pi)^{d}}$ (14)

then we obtain the equation of motion

$\mathbb{D}_{+}\left(r^{d-1}\mathbb{D}_{+}\phi\right)+\frac{\beta^{2}}{4}r^{d-1}\left(f|\mathbf{k}|^{2}-(k^{0})^{2}\right)\phi=0\
,$ (15)

where $\mathbb{D}_{+}$ is the radial derivative operator defined as

$\mathbb{D}_{+}\equiv\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\zeta}+\frac{\beta
k^{0}}{2}\ .$ (16)

Note that we use the same symbol for the field in the real domain as well as
the Fourier domain. This should not cause any confusion, since the arguments
of the field will differentiate between them.

The above second-order ordinary differential equation has two independent
solutions. The solution $G^{\rm in}(\zeta,k)$ that satisfies the boundary
conditions

$G^{\rm in}(\zeta,k)\Big{|}_{\zeta=0}=1=G^{\rm in}(\zeta,k)\Big{|}_{\zeta=1}$
(17)

is called the _ingoing boundary-to-bulk propagator_. Given this ingoing
solution, we can generate another independent solution, the _outgoing
boundary-to-bulk propagator_ , by using the time-reversal involution of the
grSK geometry, as

$G^{\rm out}(\zeta,k)=e^{-\beta k^{0}\zeta}G^{\rm in}(\zeta,-k)\ .$ (18)

With these two independent solutions, the most general solution of the Klein-
Gordon equation satisfying the boundary conditions

$\phi(\zeta,k)\Big{|}_{\zeta=0}=J_{{}_{\text{L}}}(k)\
,\quad\phi(\zeta,k)\Big{|}_{\zeta=1}=J_{{}_{\text{R}}}(k)\ ,$ (19)

takes the form

$\phi(\zeta,k)=-G^{\rm in}(\zeta,k)J_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k)+e^{\beta k^{0}}G^{\rm out}J_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k)\ .$ (20)

This form of the solution with the ingoing and the outgoing propagators was
first explored in Son:2009vu  and later detailed for the grSK geometry in
Chakrabarty:2019aeu ; Jana:2020vyx . Here
$J_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}$ and
$J_{\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}$ are the future and the past
sources defined as

$\begin{split}J_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k)&\equiv-(1+n^{{}_{\text{BE}}}_{k^{0}})J_{{}_{\text{R}}}(k)+n^{{}_{\text{BE}}}_{k^{0}}J_{{}_{\text{L}}}(k)\
,\\\ J_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k)&\equiv-n^{{}_{\text{BE}}}_{k^{0}}\left[J_{{}_{\text{R}}}(k)-J_{{}_{\text{L}}}(k)\right]\
,\end{split}$ (21)

respectively. The symbol $n^{{}_{\text{BE}}}_{k^{0}}$ denotes the Bose-
Einstein statistical factor

$n^{{}_{\text{BE}}}_{k^{0}}\equiv\frac{1}{e^{\beta k^{0}}-1}\ .$ (22)

We will also later find it convenient to write the above solution in the
right-left basis of the sources as

$\begin{split}\phi(\zeta,k)&\equiv\
g_{{}_{\text{R}}}(\zeta,k)\,J_{{}_{\text{R}}}(k)-g_{{}_{\text{L}}}(\zeta,k)\,J_{{}_{\text{L}}}(k)\
,\end{split}$ (23)

where $g_{{}_{\text{R}},{}_{\text{L}}}$ denote the boundary-to-bulk Green
functions from the right and left AdS boundaries, which in terms of the
ingoing and the outgoing Green functions take the form

$\begin{split}&g_{{}_{\text{R}}}(\zeta,k)\equiv\left(1+n^{{}_{\text{BE}}}_{k^{0}}\right)\left[G^{\text{in}}(\zeta,k)-G^{\text{out}}(\zeta,k)\right]\
,\\\ &g_{{}_{\text{L}}}(\zeta,k)\equiv
n^{{}_{\text{BE}}}_{k^{0}}\left[G^{\text{in}}(\zeta,k)-e^{\beta
k^{0}}G^{\text{out}}(\zeta,k)\right]\ .\end{split}$ (24)

###  3.2 Free Dirac theory

Next, we come to free Dirac spinor propagation in the grSK geometry. The free
Dirac action takes the form

$S_{\Psi\rm free}=\
i\oint\mathop{}\\!\mathrm{d}\zeta\int\mathop{}\\!\mathrm{d}^{d}x\sqrt{-g}\,\bar{\Psi}\Gamma^{\scalebox{0.9}{$\scriptscriptstyle
A$}}D_{\scalebox{0.9}{$\scriptscriptstyle
A$}}\Psi+i\int\mathop{}\\!\mathrm{d}^{d}x\,\bar{\Psi}\mathbb{P}_{-}\Psi\Bigg{|}_{\zeta=0}^{\zeta=1}\
.$ (25)

Notice that the first term is just the familiar massless Dirac action in
curved spacetime, where the symbol $D_{\scalebox{0.9}{$\scriptscriptstyle
A$}}$ represents the appropriate covariant derivative, and $\mathbb{P}_{\pm}$
are the appropriate projection operators. Explicitly,

$D_{\scalebox{0.9}{$\scriptscriptstyle
A$}}\equiv\partial_{\scalebox{0.9}{$\scriptscriptstyle
A$}}+\frac{1}{4}\omega_{ab\scalebox{0.9}{$\scriptscriptstyle
A$}}\Gamma^{a}\Gamma^{b}\
,\qquad\mathbb{P}_{\pm}\equiv\frac{1}{2}\left(\mathbb{I}\pm\Gamma^{(\zeta)}\right)\
.$ (26)

Here $\omega_{abA}$ are the spin connection coefficients and $\Gamma^{a}$ are
the bulk gamma matrices.555Our notation for the indices for the gamma matrices
is as follows: Lowercase Latin indices are for vielbein frame gamma matrices
whereas uppercase Latin indices are for spacetime gamma matrices. More
precisely, $\\{\Gamma^{a},\Gamma^{b}\\}=2\eta^{ab}$ and
$\\{\Gamma^{\scalebox{0.9}{$\scriptscriptstyle
A$}},\Gamma^{\scalebox{0.9}{$\scriptscriptstyle
B$}}\\}=2g^{\scalebox{0.9}{$\scriptscriptstyle
A$}\scalebox{0.9}{$\scriptscriptstyle B$}}$. When writing explicit components,
we will denote vielbein frame gamma matrices with the superscript enclosed in
parentheses. For example, $\Gamma^{(\zeta)}$. For more details about the
conventions we use, the reader is encouraged to see Loganayagam:2020eue
Explicit expressions for these can be found in Loganayagam:2020eue . The
second term in the action above is a boundary term that ensures the existence
of a well-defined variational principle. This is because spinors (as opposed
to scalars) satisfy double semi-Dirichlet boundary conditions
Henningson:1998cd ; Henneaux:1998ch .

Varying the above action with respect to $\overline{\Psi}$ gives us the Dirac
equation

$\Gamma^{\scalebox{0.9}{$\scriptscriptstyle
A$}}D_{\scalebox{0.9}{$\scriptscriptstyle A$}}\Psi=0\ .$ (27)

Passing to the Fourier domain in the boundary coordinates, as we did for the
scalar field in Eq. (14), the Dirac equation becomes a first-order ordinary
differential equation in the radial coordinate:

$\begin{split}\left\\{\frac{\mathbb{T}(\zeta)}{\sqrt{f}}\partial_{\zeta}+\mathbb{\Gamma}\left(\beta
k^{0}+\frac{1}{2}\partial_{\zeta}\ln
f\right)-\frac{\beta}{2}\Gamma^{(i)}k_{i}\right\\}\left(r^{d/2}\Psi(\zeta,k)\right)=0\
.\end{split}$ (28)

Here $\mathbb{\Gamma}$ and $\mathbb{T}$ are linear combinations of bulk gamma
matrices defined as

$\mathbb{\Gamma}\equiv\frac{1}{2}\left(\Gamma^{(\zeta)}+\Gamma^{(v)}\right)\
,\quad\mathbb{T}(\zeta)\equiv\sqrt{f}\mathbb{\Gamma}+\frac{1}{\sqrt{f}}\mathbb{\Gamma}\
.$ (29)

In the case of Dirac spinors as well, we would like to follow the same line of
logic as we did for scalars to find the solution. But here there are new
important features we have to keep in mind. First of all, the equation of
motion is now a first-order equation. Secondly, the field we are solving for
is a spinor, i.e., it has a spinorial index.

Due to the second feature, the boundary conditions of the bulk field are
slightly more complicated. This is because in general dimensions, the boundary
spinor and the bulk spinor do not have the same number of components. If
$\psi_{{}_{\text{R}}}$ and $\psi_{{}_{\text{L}}}$ are spinors on the right and
the left boundary respectively, the boundary conditions on the bulk spinor
$\Psi$ take the form

$\lim_{\zeta\to
0}r^{\frac{d}{2}}\mathbb{P}_{+}S_{{}_{0}}\Psi(\zeta,k)=\mathbb{P}_{+}S_{{}_{0}}\psi_{{}_{\text{L}}}(k)\
,\qquad\qquad\lim_{\zeta\to
1}r^{\frac{d}{2}}\mathbb{P}_{+}S_{{}_{0}}\Psi(\zeta,k)=\mathbb{P}_{+}S_{{}_{0}}\psi_{{}_{\text{R}}}(k)\
.$ (30)

Here $S_{0}$ is a constant matrix defined in Loganayagam:2020eue . Notice that
since the Dirac equation is first order in derivatives, the double Dirichlet
boundary conditions can be imposed only after projecting with an appropriate
projection matrix, which in the present case is $\mathbb{P}_{{}_{+}}$.

We can now proceed as in the scalar case by first introducing the ingoing
boundary-to-bulk propagator $S^{\rm in}$, which is a solution of the Dirac
equation and satisfies the boundary conditions

$\lim_{\zeta\to
0}r^{\frac{d}{2}}\mathbb{P}_{+}S^{\text{in}}(\zeta,k)=\mathbb{P}_{+}S_{{}_{0}}=\lim_{\zeta\to
1}r^{\frac{d}{2}}\mathbb{P}_{+}S^{\text{in}}(\zeta,k)\ .$ (31)

Given the ingoing propagator, once again the time-reversal involution of the
grSK geometry produces another solution $S^{\rm out}$ given by

$S^{\rm out}(\zeta,k)\equiv e^{-\beta k^{0}\zeta}\mathbb{T}(\zeta)S^{\rm
in}(\zeta,-k)\ .$ (32)

The most general solution of the Dirac equation satisfying the boundary
conditions in Eq.(31) can then be written as

$\Psi(\zeta,k)=\
-S^{\text{in}}(\zeta,k)\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k)-e^{\beta
k^{0}}S^{\text{out}}(\zeta,k)\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k)\ ,$ (33)

where $\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}(k)$ and
$\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}(k)$ are the boundary
sources in the past-future basis defined as

$\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}(k)\equiv\
n^{{}_{\text{FD}}}_{k^{0}}\left[\psi_{{}_{\text{R}}}(k)-\psi_{{}_{\text{L}}}(k)\right]-\psi_{{}_{\text{R}}}(k)\
,\qquad\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}(k)\equiv\
n^{{}_{\text{FD}}}_{k^{0}}\left[\psi_{{}_{\text{R}}}(k)-\psi_{{}_{\text{L}}}(k)\right]\
.$ (34)

We will also find it useful to recast the above solution in the right-left
basis of the boundary sources:

$\Psi(\zeta,k)\equiv s_{{}_{\text{R}}}(\zeta,k)\
\psi_{{}_{\text{R}}}(k)-s_{{}_{\text{L}}}(\zeta,k)\ \psi_{{}_{\text{L}}}(k)\
,$ (35)

where $s_{{}_{\text{R}}}$ and $s_{{}_{\text{L}}}$ denote the boundary-to-bulk
propagators from the right and the left AdS boundaries. In terms of the
ingoing and the outgoing propagators, these take the form

$\begin{split}&s_{{}_{\text{R}}}(\zeta,k)\equiv\
\left(1-n^{{}_{\text{FD}}}_{k^{0}}\right)\left[S^{\text{in}}(\zeta,k)-S^{\text{out}}(\zeta,k)\right]\
,\\\ &s_{{}_{\text{L}}}(\zeta,k)\equiv\
-n^{{}_{\text{FD}}}_{k^{0}}S^{\text{in}}(\zeta,k)-\left(1-n^{{}_{\text{FD}}}_{k^{0}}\right)S^{\text{out}}(\zeta,k)\
.\end{split}$ (36)

The Dirac conjugate solutions can also be arranged in the same form as in
Eq.(33). Thus, the conjugate solutions have a very similar structure as those
given above. We refer the reader to the Appendix. (A) for more details.

###  3.3 Yukawa theory

We now have all the basic setup and notation we require to discuss Yukawa
theory in the grSK spacetime. The action we consider is the sum of the free
actions we have described in the previous subsections as well as the Yukawa
interaction term.

$S=S_{\phi\rm free}+S_{\Psi\rm
free}-i\oint\mathop{}\\!\mathrm{d}\zeta\int\mathop{}\\!\mathrm{d}^{d}x\sqrt{-g}\
\lambda\phi\bar{\Psi}\Psi\ .$ (37)

Varying the above action gives Klein-Gordon and Dirac equations that are
sourced by terms linear in the Yukawa coupling:

$\frac{1}{\sqrt{-g}}\partial_{\scalebox{0.9}{$\scriptscriptstyle
B$}}\left(\sqrt{-g}g^{\scalebox{0.9}{$\scriptscriptstyle
A$}\scalebox{0.9}{$\scriptscriptstyle
B$}}\partial_{\scalebox{0.9}{$\scriptscriptstyle
A$}}\phi\right)=i\lambda\bar{\Psi}\Psi\
\qquad\text{and}\qquad\Gamma^{\scalebox{0.9}{$\scriptscriptstyle
A$}}D_{\scalebox{0.9}{$\scriptscriptstyle A$}}\Psi=\lambda\phi\Psi\ .$ (38)

These equations can be solved perturbatively in the Yukawa coupling $\lambda$.
In other words, we can express the solutions as

$\Psi=\Psi_{(0)}+\lambda\Psi_{(1)}+\lambda^{2}\Psi_{(2)}+\ldots\ ,$ (39)
$\phi=\phi_{(0)}+\lambda\phi_{(1)}+\lambda^{2}\phi_{(2)}+\ldots\ .$ (40)

The leading-order solutions $\Psi_{(0)}$ and $\phi_{(0)}$ are just the free
theory solutions that we have seen in detail in the previous subsections. We
will take these solutions to satisfy the GKPW boundary conditions we require.
Thus the higher-order terms in the solutions will be normalisable on both the
boundaries. To obtain the higher-order solutions, we will now need to develop
a technique to solve sourced Klein-Gordon and Dirac equations.

We will attack this problem with the technique of Green functions. First we
define the binormalisable bulk-to-bulk propagators. These are objects that are
normalisable on both the boundaries of the grSK spacetime and satisfy the
equations of motion with a Dirac delta source in the bulk. More precisely, the
scalar binormalisable propagator $\mathbb{G}(\zeta|\zeta_{0},k)$ satisfies

$\mathbb{D}_{+}\left(r^{d-1}\mathbb{D}_{+}\mathbb{G}(\zeta|\zeta_{0},k)\right)+\frac{\beta^{2}}{4}r^{d-1}\left(f|\mathbf{k}|^{2}-(k^{0})^{2}\right)\mathbb{G}(\zeta|\zeta_{0},k)=\frac{i\beta}{2}\delta(\zeta-\zeta_{0})\
,$ (41)

where $\delta(\zeta-\zeta_{0})$ denotes the radial delta function on the grSK
contour. The above equation must be solved with the boundary conditions

$\lim_{\zeta\to 0}\mathbb{G}(\zeta|\zeta_{0},k)=\lim_{\zeta\to
1}\mathbb{G}(\zeta|\zeta_{0},k)=0\ .$ (42)

Similarly, the spinor bulk-to-bulk propagator $\mathbb{S}(\zeta|\zeta_{0},k)$
is a square matrix in the bulk spinor space that satisfies

$\begin{split}\left\\{\frac{\mathbb{T}(\zeta)}{\sqrt{f}}\partial_{\zeta}+\mathbb{\Gamma}\left(\beta
k^{0}+\frac{1}{2}\partial_{\zeta}\ln
f\right)-\frac{\beta}{2}\Gamma^{(i)}k_{i}\right\\}\left(r^{d/2}\mathbb{S}(\zeta|\zeta_{0},k)\right)=\mathbb{1}\frac{\delta(\zeta-\zeta_{0})}{r^{d/2}f}\
,\end{split}$ (43)

where $\mathbb{1}$ is the identity in the bulk spinor space, along with the
boundary conditions

$\lim_{\zeta\to
0}r^{\frac{d}{2}}\mathbb{P}_{+}S_{{}_{0}}\mathbb{S}(\zeta|\zeta_{0},k)=0\
,\qquad\qquad\lim_{\zeta\to
1}r^{\frac{d}{2}}\mathbb{P}_{+}S_{{}_{0}}\mathbb{S}(\zeta|\zeta_{0},k)=0\ .$
(44)

An explicit derivation of the scalar bulk-to-bulk propagator can be found in
GodwinLoga . As for the spinor bulk-to-bulk propagator, explicit expressions
in terms of the boundary-to-bulk propagators can be found in Appendix.(B),
where we derive the propagators along similar lines as in Faulkner:2013bna ,
using the technique of Wronskians.

With the bulk-to-bulk propagators in our hands, we can immediately write down
expressions for the higher-order terms in the perturbative expansion of the
solutions. For all $n>0$, we have

$\Psi_{(n)}(\zeta,k)=\oint_{\zeta}\
\mathbb{S}(\zeta|\zeta_{0},k)\mathfrak{J}_{(n)}(\zeta_{0},k)\
,\quad\text{and}\quad\phi_{(n)}(\zeta,k)=\oint_{\zeta}\
\mathbb{G}(\zeta|\zeta_{0},k)\mathbb{J}_{(n)}(\zeta_{0},k)\ ,$ (45)

where $\mathfrak{J}_{(n)}(\zeta_{0},k)$ and $\mathbb{J}_{(n)}(\zeta_{0},k)$
are sources for the $n$th order term in the solutions. These can be read off
from the equations of motion. For example, at first order we have

$\begin{split}\mathfrak{J}_{(1)}(\zeta_{0},k)&\equiv\int_{k_{1}}\int_{k_{2}}(2\pi)^{d}\delta^{(d)}(k_{1}+k_{2}-k)\phi_{(0)}(\zeta_{0},k_{1})\Psi_{(0)}(\zeta_{0},k_{2})\
,\\\ \mathbb{J}_{(1)}(\zeta_{0},k)&\equiv
i\int_{k_{1}}\int_{k_{2}}(2\pi)^{d}\delta^{(d)}(k_{1}+k_{2}-k)\overline{\Psi}_{(0)}(\zeta_{0},k_{1})\Psi_{(0)}(\zeta_{0},k_{2})\
.\end{split}$ (46)

Similarly, at second-order,

$\begin{split}\mathfrak{J}_{(2)}(\zeta_{0},k)&\equiv\int_{k_{1}}\int_{k_{2}}(2\pi)^{d}\delta^{(d)}(k_{1}+k_{2}-k)\left[\phi_{(0)}(\zeta_{0},k_{1})\Psi_{(1)}(\zeta_{0},k_{2})+\phi_{(1)}(\zeta_{0},k_{1})\Psi_{(0)}(\zeta_{0},k_{2})\right]\
,\\\ \mathbb{J}_{(2)}(\zeta_{0},k)&\equiv
i\int_{k_{1}}\int_{k_{2}}(2\pi)^{d}\delta^{(d)}(k_{1}+k_{2}-k)\left[\overline{\Psi}_{(0)}(\zeta_{0},k_{1})\Psi_{(1)}(\zeta_{0},k_{2})+\overline{\Psi}_{(1)}(\zeta_{0},k_{1})\Psi_{(0)}(\zeta_{0},k_{2})\right]\
.\end{split}$ (47)

##  4 Yukawa theory and exterior EFT from grSK

We now come to the influence phase for Yukawa theory. In the previous section,
we have detailed how to obtain the solutions to the bulk equations of motion
to arbitrary orders in the Yukawa coupling constant. Substituting these
solutions into the action will then give us the tree-level influence phase up
to any desired order.

We will arrange this expansion with the notation

$S_{\text{on-shell}}=S_{(2)}+\lambda
S_{(3)}+\lambda^{2}S_{(4)}+\lambda^{3}S_{(5)}+\ldots\ ,$ (48)

where $S_{\text{on-shell}}$ is the full on-shell action. On the RHS, we have
employed a notation wherein the subscripts indicate the number of boundary
sources in the term. Thus, for example, $S_{(2)}$ is quadratic in the boundary
sources and takes the form given in Eq. (133).

The term linear in the coupling $\lambda$ is also cubic in the boundary
sources and takes the form

$S_{(3)}=-i\int_{k_{1,2,3}}\oint_{\zeta}\overline{\Psi}_{(0)}(\zeta,k_{1})\phi_{(0)}(\zeta,k_{2})\Psi_{(0)}(\zeta,k_{3})\
.$ (49)

An explicit derivation of this can be found in Appendix D. Here we have
introduced a useful notation, that we will employ throughout the rest of this
note,

$\int_{k_{1,2,\ldots,n}}\equiv\int\frac{\mathop{}\\!\mathrm{d}^{d}k_{1}}{(2\pi)^{d})}\int\frac{\mathop{}\\!\mathrm{d}^{d}k_{2}}{(2\pi)^{d})}\ldots\int\frac{\mathop{}\\!\mathrm{d}^{d}k_{n}}{(2\pi)^{d})}(2\pi)^{d}\delta^{d}(k_{1}+k_{2}+\ldots+k_{n})\
.$ (50)

The reader will notice that the three-point influence phase can be directly
obtained from the free solutions in the usual Witten-diagram manner with the
diagrams drawn on the full grSK contour. We will soon perform the monodromy
integrals and see how this answer can be arranged in terms of Witten diagrams
on one copy of the spacetime (instead of two, as in the grSK prescription).

The next correction, i.e., the term quadratic in the interaction parameter
$\lambda$, which is quartic in the boundary sources, takes the form

$\begin{split}S_{(4)}=\
&\frac{1}{2}\int_{k_{1,2,3,4}}\oint_{\zeta_{1}}\oint_{\zeta_{2}}\overline{\Psi}_{(0)}(\zeta_{2},k_{1})\Psi_{(0)}(\zeta_{2},k_{2})\mathbb{G}(\zeta_{2}|\zeta_{1},k_{3}+k_{4})\overline{\Psi}_{(0)}(\zeta_{1},k_{3})\Psi_{(0)}(\zeta_{1},k_{4})\\\
&-i\int_{k_{1,2,3,4}}\oint_{\zeta_{1}}\oint_{\zeta_{2}}\phi_{(0)}(\zeta_{2},k_{1})\overline{\Psi}_{(0)}(\zeta_{2},k_{2})\mathbb{S}(\zeta_{2}|\zeta_{1},k_{3}+k_{4})\phi_{(0)}(\zeta_{1},k_{3})\Psi_{(0)}(\zeta_{1},k_{4})\
.\end{split}$ (51)

Once again, an explicit derivation of the above results can be found in
Appendix D.

We have seen above the three-point and the four-point influence phase written
as integrals over the full grSK contour. We will now perform these monodromy
integrals. To this end, let us focus on one particular term in the three-point
influence phase. It will be clear from this discussion how the other terms can
be similarly computed. We focus on the term with two future sources and one
past source, each of which can be a scalar, a spinor, or its conjugate. Thus
we expect three terms at the end. Starting from Eq. (49) and extracting the
desired term, we have

$\begin{split}S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}&=-i\int_{k_{1,2,3}}\oint_{\zeta}\Bigg{\\{}\left[-\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k_{1})\overline{S}^{\text{in}}(\zeta,k_{1})\right]\left[-G^{\text{in}}(\zeta,k_{2})J_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k_{2})\right]\left[-e^{\beta
k^{0}_{3}}S^{\text{out}}(\zeta,k_{3})\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k_{3})\right]\\\ &\hskip
56.9055pt+\left[-\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k_{1})\overline{S}^{\text{in}}(\zeta,k_{1})\right]\left[e^{\beta
k^{0}_{2}}G^{\text{out}}(\zeta,k_{2})J_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k_{2})\right]\left[-S^{\text{in}}(\zeta,k_{3})\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k_{3})\right]\\\ &\hskip 56.9055pt+\left[-e^{\beta
k^{0}_{1}}\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k_{1})\overline{S}^{\text{out}}(\zeta,k_{1})\right]\left[-G^{\text{in}}(\zeta,k_{2})J_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k_{2})\right]\left[-S^{\text{in}}(\zeta,k_{3})\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k_{3})\right]\Bigg{\\}}\ .\end{split}$ (52)

The ingoing propagator is analytic on the grSK contour and simply goes for a
ride in the monodromy integral. The only non-analyticities in the above
expression come from the outgoing propagators. These come from the factors of
$\sqrt{f}$ and $e^{-\beta k^{0}\zeta}$. Noting that $\sqrt{f}$ flips in sign
while going from one branch of the contour to the other, and $\zeta$ jumps by
unity, we get

$\oint_{\zeta}e^{-\beta k^{0}\zeta}f^{j/2}\mathscr{F}(\zeta)=\int_{\rm
ext}e^{-\beta
k^{0}\zeta}f^{j/2}\mathscr{F}(\zeta)\times\begin{cases}\frac{-1}{1+n^{{}_{\text{BE}}}_{k^{0}}}&\text{for}\
j=0\\\ \frac{-1}{1-n^{{}_{\text{FD}}}_{k^{0}}}&\text{for}\ j=1\end{cases}\ ,$
(53)

where $\mathcal{F}(\zeta)$ is an analytic function and we have chosen to write
the integral over the left branch of the grSK contour with $\int_{\rm ext}$
defined as

$\int_{\rm ext}\equiv\int_{r_{h}}^{r_{c}}\mathop{}\\!\mathrm{d}r\ r^{d-1}\ .$
(54)

Using this result, we obtain

$\begin{split}S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}&=-i\int_{k_{1,2,3}}\int_{\rm
ext}\bigg{\\{}\frac{1}{n^{{}_{\text{FD}}}_{k_{3}^{0}}}\left[\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k_{1})\overline{S}^{\text{in}}(\zeta,k_{1})\right]\left[G^{\text{in}}(\zeta,k_{2})J_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k_{2})\right]\left[S^{\text{out}}(\zeta,k_{3})\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k_{3})\right]\\\ &\hskip
79.6678pt-\frac{1}{n^{{}_{\text{BE}}}_{k_{2}^{0}}}\left[\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k_{1})\overline{S}^{\text{in}}(\zeta,k_{1})\right]\left[G^{\text{out}}(\zeta,k_{2})J_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k_{2})\right]\left[S^{\text{in}}(\zeta,k_{3})\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k_{3})\right]\\\ &\hskip
79.6678pt+\frac{1}{n^{{}_{\text{FD}}}_{k_{1}^{0}}}\left[\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k_{1})\overline{S}^{\text{out}}(\zeta,k_{1})\right]\left[G^{\text{in}}(\zeta,k_{2})J_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k_{2})\right]\left[S^{\text{in}}(\zeta,k_{3})\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k_{3})\right]\bigg{\\}}\ ,\end{split}$ (55)

Note also that none of the functions in the integrand have poles at the
horizon (which can be gleaned from the explicit expressions of the propagators
in Jana:2020vyx ; Loganayagam:2020eue ), and thus there are no horizon-
localised contributions.

We can now perform similar computations for all the other terms in the three-
point influence phase. The term with only future sources vanishes, i.e.,
$S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}=0$, which is simply the SK
collapse condition at the three-point level. This is because the integrand is
completely analytic and thus the monodromy integral vanishes. Similarly, the
term with all past sources also vanishes, i.e.,
$S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}=0$, which is the KMS
condition at the three-point level. This is because each term has an
exponential factor, all of which multiply to give unity by momentum
conservation, and a factor of $f$ which is analytic around $r=r_{h}$.
Therefore, the only other non-vanishing term in the three-point influence
phase is the term with two past sources. This term looks like

$\begin{split}S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}&=-i\int_{k_{1,2,3}}\int_{\rm
ext}\bigg{\\{}-\frac{1}{n^{{}_{\text{FD}}}_{k_{2}^{0}+k_{3}^{0}}}\left[\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k_{1})\overline{S}^{\text{in}}(\zeta,k_{1})\right]\left[G^{\text{out}}(\zeta,k_{2})J_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k_{2})\right]\left[S^{\text{out}}(\zeta,k_{3})\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k_{3})\right]\\\ &\hskip
85.35826pt+\frac{1}{n^{{}_{\text{BE}}}_{k_{1}^{0}+k_{3}^{0}}}\left[\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k_{1})\overline{S}^{\text{out}}(\zeta,k_{1})\right]\left[G^{\text{in}}(\zeta,k_{2})J_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k_{2})\right]\left[S^{\text{out}}(\zeta,k_{3})\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k_{3})\right]\\\ &\hskip
85.35826pt-\frac{1}{n^{{}_{\text{FD}}}_{k_{1}^{0}+k_{2}^{0}}}\left[\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k_{1})\overline{S}^{\text{out}}(\zeta,k_{1})\right]\left[G^{\text{out}}(\zeta,k_{2})J_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k_{2})\right]\left[S^{\text{in}}(\zeta,k_{3})\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k_{3})\right]\bigg{\\}}\ .\end{split}$ (56)

Thus the full three-point influence phase takes the form

$S_{(3)}=S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}+S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}\ .$ (57)

We can compute the higher point contributions to the influence phase using
similar arguments as above. Rather than illustrating explicit calculations, we
will introduce a series of diagrammatic Feynman rules. Through these rules,
the reader can determine the on-shell action at any desired order.

###  4.1 Feynman rules for Yukawa Theory

The diagrammatic rules for the computation of the tree-level influence phase
(on-shell action) are given below. The convention we use for the direction of
momentum flow is as follows. In any boundary-to-bulk propagator, the momentum
flows from the boundary to the bulk. In the bulk-to-bulk propagators, the
momentum flows from the left to the right.

The scalar boundary-to-bulk propagators are as follows.

$\bullet$$\zeta$$k$$\equiv-iG^{\text{in}}(\zeta,k)\
J_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}(k)\ ,$
$\bullet$$\zeta$$k$$\equiv-iG^{\rm out}(\zeta,k)\
J_{\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}(k)\ .$ (58)

The scalar bulk-to-bulk propagators are

$k$$\zeta_{1}$$\zeta_{2}$$\bullet$$\bullet$$\equiv-i\mathbb{G}_{\text{ret}}(\zeta_{2}|\zeta_{1},k)\
,$
$k$$\zeta_{1}$$\zeta_{2}$$\bullet$$\bullet$$\equiv-i\mathbb{G}_{\text{adv}}(\zeta_{2}|\zeta_{1},k)\
.$ (59)

The spinor boundary-to-bulk propagators are given below. Here the additional
arrows represent the spinorial direction ($\overline{\psi}$ to $\psi$), which
becomes the charge arrow in QED.

$\bullet$$\zeta$$k$$\uparrow$$\equiv-iS^{\text{in}}(\zeta,k)\
\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}(k)\ ,$
$\bullet$$\zeta$$k$$\uparrow$$\equiv-iS^{\text{out}}(\zeta,k)\
\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}(k)\ ,$ (60)
$\bullet$$\zeta$$k$$\downarrow$$\equiv-i\overline{S}^{\text{in}}(\zeta,k)\
\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}(k)\ ,$
$\bullet$$\zeta$$k$$\downarrow$$\equiv-i\overline{S}^{\text{out}}(\zeta,k)\
\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}(k)\ .$ (61)

The spinor bulk-to-bulk propagators are

$\rightarrow$$k$$\zeta_{1}$$\zeta_{2}$$\bullet$$\bullet$$\equiv-i\mathbb{S}_{\text{ret}}(\zeta_{2}|\zeta_{1},k)\
,$
$\rightarrow$$k$$\zeta_{1}$$\zeta_{2}$$\bullet$$\bullet$$\equiv-i\mathbb{S}_{\text{adv}}(\zeta_{2}|\zeta_{1},k)\
.$ (62)

Finally, the vertices are as follows.

$\bullet$$k_{1}$$k_{2}$$k_{3}$$=-i\lambda\frac{n^{{}_{\text{BE}}}_{-k^{0}_{3}}}{n^{{}_{\text{BE}}}_{k^{0}_{1}+k^{0}_{2}}}\
,$
$\bullet$$k_{1}$$k_{2}$$k_{3}$$=-i\lambda\frac{-n^{{}_{\text{FD}}}_{-k^{0}_{2}}n^{{}_{\text{BE}}}_{-k^{0}_{3}}}{-n^{{}_{\text{FD}}}_{k^{0}_{1}}}\
,$ (63)
$\bullet$$k_{1}$$k_{2}$$k_{3}$$=-i\lambda\frac{-n^{{}_{\text{FD}}}_{-k^{0}_{3}}}{-n^{{}_{\text{FD}}}_{k^{0}_{1}+k^{0}_{2}}}\
,$
$\bullet$$k_{1}$$k_{2}$$k_{3}$$=-i\lambda\frac{(-n^{{}_{\text{FD}}}_{-k^{0}_{2}})(-n^{{}_{\text{FD}}}_{-k^{0}_{3}})}{n^{{}_{\text{BE}}}_{k^{0}_{1}}}\
.$ (64)

In each of the above vertices, we have adopted the convention that all the
momenta flow into the vertex. The reader may note that these rules are the
same as those given in Gelis:2019yfm .

Along with the above Feynman rules, to obtain the influence phase, we must
further

  1. 1.

multiply any term with $n$ boundary sources by $(-i)^{n-1}$.

  2. 2.

multiply every past source by $(-1)$.

  3. 3.

multiply each $\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k)$ or $\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}(k)$ by
$\frac{1}{-n^{{}_{\text{FD}}}_{k^{0}}}$ and every
$J_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}(k)$ by
$\frac{1}{n^{{}_{\text{BE}}}_{k^{0}}}$.

  4. 4.

multiply every external conjugate spinor source by $i$.

With the rules above, the diagrams that contribute to the three-point
influence phase are given in Fig. (4). The reader can easily check that the
expressions in Eq. (56) and Eq. (55) are consistent with those generated by
these diagrams with the rules above. In all the diagrams below, the blue line
denotes the left AdS boundary. Here, the charge arrow of the spinors are
always taken to be flowing from the boundary and from the left to the right.

$\bullet$ (a)

$\bullet$ (b)

$\bullet$ (c)

$\bullet$ (d)

$\bullet$ (e)

$\bullet$ (f)

Figure 4: Witten diagrams that contribute to the three-point influence phase.

Similarly, the diagrams that contribute to the four-point influence phase are
as follows.

$\bullet$$\bullet$ (a)

$\bullet$$\bullet$ (b)

$\bullet$$\bullet$ (c)

$\bullet$$\bullet$ (d)

$\bullet$$\bullet$ (e)

$\bullet$$\bullet$ (f)

Figure 5: Witten diagrams that contribute to the
$S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}$ term in the four-point
exchange influence phase.

$\bullet$$\bullet$ (a)

$\bullet$$\bullet$ (b)

$\bullet$$\bullet$ (c)

$\bullet$$\bullet$ (d)

$\bullet$$\bullet$ (e)

$\bullet$$\bullet$ (f)

Figure 6: Witten diagrams that contribute to the
$S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}$ term in the four-point
exchange influence phase.

$\bullet$$\bullet$ (a)

$\bullet$$\bullet$ (b)

$\bullet$$\bullet$ (c)

Figure 7: Witten diagrams that contribute to the
$S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}$ term in the four-point
exchange influence phase.

$\bullet$$\bullet$ (a)

$\bullet$$\bullet$ (b)

$\bullet$$\bullet$ (c)

$\bullet$$\bullet$ (d)

$\bullet$$\bullet$ (e)

$\bullet$$\bullet$ (f)

$\bullet$$\bullet$ (g)

$\bullet$$\bullet$ (h)

$\bullet$$\bullet$ (i)

$\bullet$$\bullet$ (j)

$\bullet$$\bullet$ (k)

$\bullet$$\bullet$ (l)

Figure 8: Witten diagrams that contribute to the
$S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}$ term in the four-point
exchange influence phase.

##  5 Column vector representation of Yukawa influence phase

In this section, we will express the three-point and the four-point Yukawa
influence phase in a convenient notation, i.e., the column vector
representation. The column vector representation naturally gives the spectral
functions in the real-time formalism Chaudhuri:2018ihk . This is not
surprising as real-time correlators admit Käll’en-Lehmann-like spectral
representations Evans:1990hy ; Evans:1991ky . It has been very useful in
perturbative techniques, computation of transport coefficients, etc.
Henning:1993gh ; Carrington:2006xj .

The column vector representation manifestly incorporates the microscopic
unitarity and the thermality of the system as shown in Chaudhuri:2018ihk .
Here, the authors showed how thermal bosonic Out-of-Time-Ordered Correlators
(OTOCs) can be organised in spectral representations using column vectors.
While for scalar holographic theories, it has been shown that the same results
can be obtained from the bulk GodwinLoga , the generalisation to include
spinors has not been done so far. From a boundary perspective, we anticipate
the existence of a similar representation for fermions. Here, we aim to
demonstrate that this expectation is indeed supported by the bulk side as
well.

We now turn our attention to the explicit construction of the column vector
representation. In this representation, boundary scalar sources are expressed
as

$\begin{split}&J_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k)\equiv\mathbf{J}(k)\cdot e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k})\ ,\qquad J_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k)\equiv\mathbf{J}(k)\cdot e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k})\ ,\end{split}$ (65)

where we have employed the notation $\bar{k}\equiv-k$ to avoid clutter, and
the symbol $\mathbf{J}(k)$ denotes the array of boundary scalar sources:

$\mathbf{J}(k)\equiv\begin{pmatrix}J_{{}_{\text{R}}}(k)&\quad-
J_{{}_{\text{L}}}(k)\end{pmatrix}\ .$ (66)

Furthermore, $e_{\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}(k)$ and
$e_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}(k)$ are column vectors of
the form

$e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(k)\equiv\begin{pmatrix}1+n^{{}_{\text{BE}}}_{k^{0}}\\\
1+n^{{}_{\text{BE}}}_{k^{0}}\end{pmatrix}\ ,\quad\text{and}\quad
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(k)\equiv\begin{pmatrix}n^{{}_{\text{BE}}}_{k^{0}}\\\
1+n^{{}_{\text{BE}}}_{k^{0}}\end{pmatrix}\ .$ (67)

Similarly, we now introduce the column vector representation for spinors. In
column vector notation, the boundary spinor sources are

$\begin{split}&\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k)\equiv\boldsymbol{\psi}(k)\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{{}_{(1)}}(\bar{k})\
,\qquad\psi_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k)\equiv\boldsymbol{\psi}(k)\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}^{{}_{(1)}}(\bar{k})\ ,\\\
&\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k)\equiv\boldsymbol{\overline{\psi}}(k)\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{{}_{(1)}}(\bar{k})\
,\qquad\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k)\equiv\boldsymbol{\overline{\psi}}(k)\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}^{{}_{(1)}}(\bar{k})\
,\end{split}$ (68)

where $\boldsymbol{\psi}(k)$ and $\boldsymbol{\overline{\psi}}(k)$ are arrays
of boundary spinor sources defined as

$\boldsymbol{\psi}(k)\equiv\begin{pmatrix}\psi_{{}_{\text{R}}}(k)&\quad-\psi_{{}_{\text{L}}}(k)\end{pmatrix}\
,\qquad\boldsymbol{\overline{\psi}}(k)\equiv\begin{pmatrix}\overline{\psi}_{{}_{\text{R}}}(k)&\quad-\overline{\psi}_{{}_{\text{L}}}(k)\end{pmatrix}\
.$ (69)

Here again, $f_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{{}_{(1)}}$ and
$f_{\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}^{{}_{(1)}}$ are two-
dimensional column vectors whose explicit expressions are

$f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(k)\equiv\begin{pmatrix}1-n^{{}_{\text{FD}}}_{k^{0}}\\\
1-n^{{}_{\text{FD}}}_{k^{0}}\end{pmatrix}\ ,\quad
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(k)\equiv\begin{pmatrix}-n^{{}_{\text{FD}}}_{k^{0}}\\\
1-n^{{}_{\text{FD}}}_{k^{0}}\end{pmatrix}\ .$ (70)

For convenience, we also define a second set of column vectors as

$\begin{split}e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(k)&\equiv e^{-\beta
k^{0}}e_{\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}^{{}_{(1)}}(k)\ ,\qquad
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{{}_{(2)}}(k)\equiv e^{-\beta
k^{0}}e_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{{}_{(1)}}(k)\ ,\\\
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(k)&\equiv-e^{-\beta
k^{0}}f_{\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}^{{}_{(1)}}(k)\ ,\qquad
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{{}_{(2)}}(k)\equiv-e^{-\beta
k^{0}}f_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{{}_{(1)}}(k)\
.\end{split}$ (71)

Note that the column vectors for spinors can be obtained from the scalar
column vectors by replacing the Bose-Einstein factor
$n^{{}_{\text{BE}}}_{k^{0}}$ with the negative of the Fermi-Dirac factor
$-n^{{}_{\text{FD}}}_{k^{0}}$, or equivalently every $e^{-\beta k^{0}}$ by
$-e^{-\beta k^{0}}$.

### Three-point Yukawa Influence Phase

We will now present the three-point Yukawa influence phase in Eq. (55) and Eq.
(56), in the column vector representation:

$\begin{split}&S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}=i\int_{k_{1,2,3}}\Bigg{\\{}\langle\overline{S}^{\rm in}G^{\rm in}S^{\rm
out}\rangle\cdot f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{3})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{3})\right]\\\ &\hskip
71.13188pt+\langle\overline{S}^{\rm in}G^{\rm out}S^{\rm in}\rangle\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes\left[e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{2})-e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{2})\right]\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{{}_{(1)}}(\bar{k}_{3})\\\
&\hskip 71.13188pt+\langle\overline{S}^{\rm out}G^{\rm in}S^{\rm
in}\rangle\cdot\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{1})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{1})\right]\otimes
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{3})\Bigg{\\}}\ ,\end{split}$ (72)

and

$\begin{split}&S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}=-i\int_{k_{1,2,3}}\Bigg{\\{}\langle\overline{S}^{\rm in}G^{\rm
out}S^{\rm out}\rangle\cdot f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes\left[e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{2})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{3})-e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{3})\right]\\\ &\hskip
71.13188pt+\langle\overline{S}^{\rm out}G^{\rm in}S^{\rm
out}\rangle\cdot\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{3})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{3})\right]\otimes
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{{}_{(1)}}(\bar{k}_{2})\\\
&\hskip 71.13188pt+\langle\overline{S}^{\rm out}G^{\rm out}S^{\rm
in}\rangle\cdot\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{1})\otimes
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{2})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{2})\right]\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{3})\Bigg{\\}}\ .\end{split}$ (73)

For notational convenience, we have used the definition666Here,
$\scalebox{1.2}{$\scriptscriptstyle\bbalpha$},\scalebox{1.2}{$\scriptscriptstyle\bbbeta$},\ldots$
and $\alpha,\beta,\ldots$ denote the bulk and boundary spinor indices
respectively.

$\langle\overline{S}^{x_{1}}G^{x_{2}}S^{x_{3}}\rangle\equiv\int_{\rm
ext}\left[\overline{S}^{x_{1}}(\zeta,k_{1})\right]^{\alpha}_{\
\scalebox{1.2}{$\scriptscriptstyle\bbalpha$}}G^{x_{2}}(\zeta,k_{2})\left[S^{x_{3}}(\zeta,k_{3})\right]^{\scalebox{1.2}{$\scriptscriptstyle\bbalpha$}}_{\
\beta}\left[\boldsymbol{\overline{\psi}}(k_{1})\right]_{\alpha}\otimes\mathbf{J}(k_{2})\otimes\left[\boldsymbol{\psi}(k_{3})\right]^{\beta}\
,$ (74)

where $x_{1},x_{2},x_{3}\in\\{\text{in},\text{out}\\}$. Notice that all the
statistical factors are absorbed into the column vectors. We will see that
something similar continues to happen even at four-point, thus proving the
usefulness of the column vector representation.

### Four-point Yukawa Influence Phase

Let’s now come to the four-point Yukawa influence phase at the tree level.
Here, we will find it convenient to classify all the terms and diagrams into
two sets, one of which has a bulk scalar exchange and the other a bulk spinor
exchange. Thus, we will write

$S_{(4)}=S_{4,\mathbb{G}}+S_{4,\mathbb{S}}\ ,$ (75)

where the $S_{4,\mathbb{G}}$ and the $S_{4,\mathbb{S}}$ denote the terms with
the scalar and the spinor bulk-to-bulk propagators respectively. Furthermore,
we will decompose each of these terms according to combinations of boundary
sources in them. That is,

$\begin{split}S_{4,\mathbb{G}}&=S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}},\mathbb{G}}+S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}},\mathbb{G}}+S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}},\mathbb{G}}+S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle P$}},\mathbb{G}}\ ,\\\
S_{4,\mathbb{S}}&=S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}},\mathbb{S}}+S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}},\mathbb{S}}+S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}},\mathbb{S}}+S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle P$}},\mathbb{S}}\ ,\end{split}$
(76)

where the subscripts denote the number of different boundary sources. Again
(similar to the three-point influence phase), the terms with only future or
past sources, vanish, i.e., $S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}=0=S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}$. This means the SK collapse
and the KMS condition also hold at the four-point tree level.

Here, we will not give all the terms of the four-point Yukawa influence phase,
but only explicitly write down the $S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle P$}},\mathbb{G}}$ term. The
interested reader is referred to Appendix (E) for the full set of terms.

$\begin{split}&S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}},\mathbb{G}}=-\int_{k_{1,2,3,4}}\Big{\\{}\langle\overline{S}^{\text{in}}S^{\text{in}}\mathbb{G}_{\text{adv}}\overline{S}^{\text{in}}S^{\text{out}}\rangle\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{3})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{4})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{4})\right]\\\ &\hskip
85.35826pt+\langle\overline{S}^{\text{in}}S^{\text{in}}\mathbb{G}_{\text{adv}}\overline{S}^{\text{out}}S^{\text{in}}\rangle\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{3})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{3})\right]\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{4})\Big{\\}}\ ,\end{split}$ (77)

where we have defined

$\begin{split}\langle\overline{S}^{x_{1}}S^{x_{2}}&\mathbb{G}_{x}\overline{S}^{x_{3}}S^{x_{4}}\rangle\equiv\int_{\text{ext}_{1}}\int_{\text{ext}_{2}}\left[\overline{S}^{x_{1}}(\zeta_{2},k_{1})\right]^{\alpha}_{\
\scalebox{1.2}{$\scriptscriptstyle\bbalpha$}}\left[S^{x_{2}}(\zeta_{2},k_{2})\right]^{\scalebox{1.2}{$\scriptscriptstyle\bbalpha$}}_{\
\beta}\mathbb{G}_{x}(\zeta_{2}|\zeta_{1},k_{3}+k_{4})\\\ &\hskip
85.35826pt\times\left[\overline{S}^{x_{3}}(\zeta_{1},k_{3})\right]^{\gamma}_{\
\scalebox{1.2}{$\scriptscriptstyle\bbbeta$}}\left[S^{x_{4}}(\zeta_{1},k_{4})\right]^{\scalebox{1.2}{$\scriptscriptstyle\bbbeta$}}_{\
\delta}\left[\boldsymbol{\overline{\psi}}(k_{1})\right]_{\alpha}\otimes\left[\boldsymbol{\psi}(k_{2})\right]^{\beta}\otimes\left[\boldsymbol{\overline{\psi}}(k_{3})\right]_{\gamma}\otimes\left[\boldsymbol{\psi}(k_{4})\right]^{\delta}\
.\end{split}$ (78)

where $x_{1},x_{2},x_{3}\in\\{\text{in},\text{out}\\}$ and
$x\in\\{\text{ret},\text{adv}\\}$ . Notice once again, as in the case of the
three-point functions, that all the statistical factors are naturally
incorporated into the column vectors.

##  6 Discussion

In this work, we have constructed an open effective field theory for a finite-
temperature holographic system comprising interacting fermions. This was
achieved by employing the gravitational dual of the Schwinger-Keldysh
formalism. While this formalism had been previously used for free fermions
Loganayagam:2020eue ; Loganayagam:2020iol , our contribution lies in its
application to interacting fermions for the first time.

In the bulk, our framework derives an exterior effective field theory in a
black hole background. The setup in the bulk involves a massless scalar and a
massless Dirac field with a Yukawa interaction in the AdS black hole
background. Additionally, we have developed Witten diagrammatics for this
exterior effective field theory which allows us to write down the Schwinger-
Keldysh correlators of the boundary theory. We have provided explicit
expressions for these correlators up to four-point functions, at tree level in
the bulk, using Witten diagrams. However, the Witten diagrammatics we provide
can be used to compute the tree-level influence phase to arbitrary orders in
the Yukawa coupling constant.

Another contribution of our work is the derivation of the column vector
representation for fermions. This representation facilitates the concise
expression of correlators wherein the Kubo-Martin-Schwinger (KMS) conditions
and the Schwinger-Keldysh collapse conditions become manifest.

This work has many clear future directions. One of them is to understand
quantum electrodynamics (QED) in the gravitational Schwinger-Keldysh geometry.
Now that we understand how to deal with fermions interacting with scalars,
this will be the next natural step. The gauge field of QED can be decomposed
into (designer) scalars as in Ghosh:2020lel , and the QED vertex will reduce
to interactions between these scalars and the fermions. Thus, it will be
interesting to see how our work generalises to the case of designer scalars,
and thus gauge fields. We intend to explore this question in the future.

Physically, in the bulk, this corresponds to the question of photons and
matter scattering against each other in a black hole background. More
precisely, how does the Hawking radiation of the black hole, comprising
photons and matter, affect the scattering of particles thrown in from the
boundary? This question becomes particularly interesting in the case of
charged black holes Loganayagam:2020iol , since now the black hole couples to
fields not only gravitationally but also electromagnetically.

Another future direction is to study the divergences present in the many bulk
integrals in this work. An interesting question would be to ask whether the
prescription of source-renormalisation, first detailed in Jana:2020vyx ,
becomes important in the present case. This will be an independent check of
the source-renormalisation prescription.

Throughout this work, we have restricted ourselves to computing the bulk on-
shell action, which is equivalent to the tree-level influence phase. It would
be interesting to explore the loop contributions in the bulk. These loop
corrections correspond to the large $N$ corrections in the boundary field
theory.

It is also interesting to ask how much of the analysis presented here
generalises to the case of de Sitter spacetime. In the spirit of
Sleight:2021plv ; Schaub:2023scu ; Bhattacharya:2023twz , we ask what our
analysis teaches us about cosmological correlators in the Schwinger-Keldysh
formalism.

## Acknowledgements

We express our sincere gratitude to Subhro Bhattacharjee, R. Loganayagam,
Gautam Mandal, Suvrat Raju, Mukund Rangamani, Tamoghna Ray, Omkar Shetye,
Akhil Sivakumar, and the ICTS string group for valuable discussions. We
acknowledge the support of the Department of Atomic Energy, Government of
India, under project no. RTI4001, and the unwavering and generous support
provided by the people of India towards research in the fundamental sciences.

##  Appendix A Conjugate Dirac equation and its propagators

In this appendix, we will discuss the conjugate Dirac equation and its
solutions. Varying the Dirac action (25) with respect to the field $\Psi$
gives us the conjugate Dirac equation:

$\left(r^{d/2}\overline{\Psi}(\zeta,k)\right)\left\\{\overleftarrow{\partial}_{\zeta}\frac{\mathbb{T}(\zeta)}{\sqrt{f}}+\mathbb{\Gamma}\left(\beta
k^{0}+\frac{1}{2}\partial_{\zeta}\ln
f\right)-\frac{\beta}{2}\Gamma^{(i)}k_{i}\right\\}=0\ ,$ (79)

where we have already passed to the Fourier domain (in the boundary
coordinates) by writing the conjugate field as

$\overline{\Psi}(\zeta,x)\equiv\int_{k}e^{ikx}\ \overline{\Psi}(\zeta,k)\ .$
(80)

Here the leftward pointing arrow on the radial derivative denotes that the
derivative acts on the object that is to its left. Note that this conjugate
equation (and its solutions) can also be derived by performing the conjugation
operation on the Dirac equation, i.e., by using
$\overline{\Psi}\equiv\Psi^{\dagger}\Gamma^{(v)}$. But here we will treat it
as an independent equation and solve it accordingly.

The boundary conditions for the conjugate field are taken to be the conjugated
version of those given for the Dirac field Eq. (30). Given the right and the
left boundary conjugate sources, $\overline{\psi}_{{}_{\text{R}}}$ and
$\overline{\psi}_{{}_{\text{L}}}$ respectively, the boundary conditions on the
bulk conjugate field $\overline{\Psi}$ take the form

$\lim_{\zeta\to
0}r^{\frac{d}{2}}\overline{\Psi}(\zeta,k)\mathbb{P}_{{}_{-}}=\overline{\psi}_{{}_{\text{L}}}(k)\overline{S}_{0}\mathbb{P}_{{}_{-}}\
,\qquad\lim_{\zeta\to
1}r^{\frac{d}{2}}\overline{\Psi}(\zeta,k)\mathbb{P}_{{}_{-}}=\overline{\psi}_{{}_{\text{R}}}(k)\overline{S}_{0}\mathbb{P}_{{}_{-}}\
,$ (81)

where we have defined

$\overline{\psi}(k)\equiv\psi^{\dagger}(k)\gamma^{(v)}\,\quad\text{and}\quad\overline{S}_{0}\equiv\left[\gamma^{(v)}\right]^{\dagger}\left[S_{0}\right]^{\dagger}\
\Gamma^{(v)}\ .$ (82)

Here $\gamma^{(v)}$ is a boundary gamma matrix.

We can now proceed with the usual procedure: find ingoing and outgoing
propagators and then build the full solution from them. The ingoing boundary-
to-bulk conjugate propagator $\overline{S}^{\text{in}}$ satisfies the boundary
conditions

$\lim_{\zeta\to
0}r^{\frac{d}{2}}\overline{S}^{\text{in}}(\zeta,k)\mathbb{P}_{{}_{-}}=\overline{S}_{0}\mathbb{P}_{{}_{-}}=\lim_{\zeta\to
1}r^{\frac{d}{2}}\overline{S}^{\text{in}}(\zeta,k)\mathbb{P}_{{}_{-}}\ .$ (83)

The time-reversal involution of the grSK geometry then produces the outgoing
boundary-to-bulk conjugate propagator $\overline{S}^{\text{out}}$, given
by777The negative sign is convenient at the level of the Feynman rules for the
Witten diagrams.

$\overline{S}^{\text{out}}(\zeta,k)=-e^{-\beta k^{0}\zeta}\
\overline{S}^{\text{in}}(\zeta,-k)\mathbb{T}(\zeta)\ .$ (84)

The boundary limits of the outgoing conjugate propagator
$\overline{S}^{\text{out}}(\zeta,k)$ defined above, can also be obtained using
Eq. (83). Now, using the ingoing and the outgoing boundary-to-bulk propagators
constructed above, the complete solution of the conjugate Dirac equation can
be written as

$\overline{\Psi}(\zeta,k)=\
-\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}(k)\overline{S}^{\text{in}}(\zeta,k)-\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}(k)\overline{S}^{\text{out}}(\zeta,k)e^{\beta k^{0}}\ .$ (85)

Here we have defined the conjugate boundary sources in the past-future basis
as

$\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}(k)\equiv
n^{{}_{\text{FD}}}_{k^{0}}\left[\overline{\psi}_{{}_{\text{R}}}(k)-\overline{\psi}_{{}_{\text{L}}}(k)\right]-\overline{\psi}_{{}_{\text{R}}}(k)\
,\quad\overline{\psi}_{\bar{\scalebox{0.9}{$\scriptscriptstyle P$}}}(k)\equiv
n^{{}_{\text{FD}}}_{k^{0}}\left[\overline{\psi}_{{}_{\text{R}}}(k)-\overline{\psi}_{{}_{\text{L}}}(k)\right]\
.$ (86)

In the right-left basis, the solution takes the form

$\overline{\Psi}(\zeta,k)=\overline{\psi}_{{}_{\text{R}}}(k)\overline{s}_{{}_{\text{R}}}(\zeta,k)-\overline{\psi}_{{}_{\text{L}}}(k)\overline{s}_{{}_{\text{L}}}(\zeta,k)\
,$ (87)

where

$\begin{split}\overline{s}_{{}_{\text{R}}}(\zeta,k)&\equiv(1-n^{{}_{\text{FD}}}_{k^{0}})\left\\{\overline{S}^{\text{in}}(\zeta,k)-\overline{S}^{\text{out}}(\zeta,k)\right\\}\
,\\\
\overline{s}_{{}_{\text{L}}}(\zeta,k)&\equiv(1-n^{{}_{\text{FD}}}_{k^{0}})\left\\{-e^{-\beta
k^{0}}\overline{S}^{\text{in}}(\zeta,k)-\overline{S}^{\text{out}}(\zeta,k)\right\\}\
.\\\ \end{split}$ (88)

##  Appendix B Bulk-to-bulk propagators in the grSK geometry

In this section, we will discuss the derivation of bulk-to-bulk propagators in
the grSK geometry. We will begin with a quick review of scalar bulk-to-bulk
propagators and how they are obtained. Here we will mostly follow the
techniques used in Arnold:2011hp ; Faulkner:2013bna  and also implemented for
the grSK geometry in GodwinLoga .

Following this discussion on scalars, we will come to a detailed derivation of
the spinor bulk-to-bulk propagator in the grSK geometry.

### Scalar bulk-to-bulk propagator

The scalar bulk-to-bulk propagator $\mathbb{G}(\zeta|\zeta_{0},k)$ normalised
to unity at the left and the right boundary (henceforth binormalisable) is
given by

$\mathbb{G}(\zeta|\zeta_{0},k)=\frac{e^{\beta
k^{0}\zeta_{0}}}{1+n^{{}_{\text{BE}}}_{k^{0}}}\frac{g_{{}_{\text{L}}}(\zeta_{>},k)g_{{}_{\text{R}}}(\zeta_{<},k)}{K^{\text{in}}(k)-K^{\text{in}}(-k)}\
.$ (89)

Here the symbols $\zeta_{<}$ and $\zeta_{>}$ are defined by

$\zeta_{<}\equiv\Bigl{\\{}\begin{array}[]{cc}\zeta&\text{if $\zeta$ comes
before $\zeta_{0}$ on the grSK contour}\\\ \zeta_{0}&\text{if $\zeta_{0}$
comes before $\zeta$ on the grSK contour}\ ,\end{array}$ (90)

and

$\zeta_{>}\equiv\Bigl{\\{}\begin{array}[]{cc}\zeta&\text{if $\zeta$ comes
after $\zeta_{0}$ on the grSK contour}\\\ \zeta_{0}&\text{if $\zeta_{0}$ comes
after $\zeta$ on the grSK contour}\ .\end{array}$ (91)

Since the bulk-to-bulk propagator has to solve the free Klein-Gordon equation
away from the sources, it is clear that it has to be proportional to the
boundary-to-bulk propagators. Left and right-normalisability then fixes this
combination of the boundary-to-bulk propagators to be
$g_{{}_{\text{L}}}(\zeta_{>},k)g_{{}_{\text{R}}}(\zeta_{<},k)$. The other
factors in the above expression must be fixed using the discontinuity
condition coming from the delta function source in the Klein-Gordon equation.
Here, $K^{\text{in}}$ is the retarded boundary two-point correlator, obtained
by taking the boundary limit of the ingoing boundary-to-bulk propagator as
shown in Ghosh:2020lel .

Apart from the binormalisable bulk-to-bulk propagator, we also require the
retarded and the advanced bulk-to-bulk propagators, given by

$\begin{split}\mathbb{G}_{\text{ret}}(\zeta|\zeta_{0},k)&=\ -\frac{e^{\beta
k^{0}\zeta_{0}}}{1+n^{{}_{\text{BE}}}_{k^{0}}}\frac{G^{\text{in}}(\zeta_{>},k)g_{{}_{\text{R}}}(\zeta_{<},k)}{K^{\text{in}}(k)-K^{\text{in}}(-k)}\
,\\\ \mathbb{G}_{\text{adv}}(\zeta|\zeta_{0},k)&=\ -\frac{e^{\beta
k^{0}\zeta_{0}}}{1+n^{{}_{\text{BE}}}_{k^{0}}}\frac{G^{\text{out}}(\zeta_{>},k)g_{{}_{\text{R}}}(\zeta_{<},k)}{K^{\text{in}}(k)-K^{\text{in}}(-k)}\
.\end{split}$ (92)

These are both propagators that are normalisable at the left boundary.
Furthermore, $\mathbb{G}_{\text{ret}}$ is analytic in the upper half-plane
(UHP) of frequency of $k^{0}$, and $\mathbb{G}_{\text{adv}}$ is analytic in
the lower half-plane (LHP) frequency of $k^{0}$. We will now present an
argument to show these analyticity properties.

We begin by detailing the analyticity properties of the ingoing and the
outgoing boundary-to-bulk propagators. The boundary-to-bulk propagators,
$G^{\text{in}}$ and $G^{\text{out}}$, following Loganayagam:2022zmq , can be
written as

$G^{\text{in}}(\zeta,k)=K^{\text{in}}(k)\widetilde{G}^{\text{in}}(\zeta,k)\
,\qquad
G^{\text{out}}(\zeta,k)=K^{\text{in}}(-k)\widetilde{G}^{\text{out}}(\zeta,k)\
,$ (93)

where $\widetilde{G}^{\text{in}}$ and $\widetilde{G}^{\text{out}}$ are
generically regular888This is easy to see in $d=2$, from the explicit
expressions of these propagators provided in Jana:2020vyx . functions of
$k^{0}$. Here we have used the intuition that the ingoing propagator should
have poles only at quasinormal frequencies Birmingham:2001pj . It also
highlights the fact that $G^{\text{in}}$ is analytic in the UHP of $k^{0}$.

Coming to the retarded and the advanced bulk-to-bulk propagators in Eq. (92),
as far as analyticity in the $k^{0}$ plane is concerned, we can replace
$\left[K^{\text{in}}(k)-K^{\text{in}}(-k)\right]$ by
$\left[K^{\text{in}}(k)K^{\text{in}}(-k)\right]$. This leads to

$\mathbb{G}_{\text{ret}}(k)=K^{\text{in}}(k)\widetilde{\mathbb{G}}_{\text{ret}}(k)\
,\qquad\mathbb{G}_{\text{adv}}(k)=K^{\text{in}}(-k)\widetilde{\mathbb{G}}_{\text{adv}}(k)\
,$ (94)

where $\widetilde{\mathbb{G}}_{\text{ret}}$ and
$\widetilde{\mathbb{G}}_{\text{adv}}$ are again generically regular functions
of $k^{0}$. Therefore $\mathbb{G}_{\text{ret}}$ and $\mathbb{G}_{\text{adv}}$
are analytic in the upper and the lower halves of the $k^{0}$ plane
respectively.

### Spinor bulk-to-bulk propagators

We now come to the spinor bulk-to-bulk propagators. The computation of the
spinor bulk-to-bulk propagator will require the evaluation of a quantity
$\mathfrak{W}(\zeta,k)$ which satisfies
$\partial_{\zeta}\mathfrak{W}(\zeta,k)=0$, i.e., the analogue of the Wronskian
of a second-order ODE. We will, therefore, spend some time in computing this
quantity explicitly and then use it to compute the spinor bulk-to-bulk
propagator.

#### Derivation of the conserved quantity

Consider $\overline{\Psi}_{2}$ and $\Psi_{1}$, which solve the conjugate Dirac
equation and the Dirac equation, given by

$\left(r^{d/2}\overline{\Psi}_{2}(\zeta,k)\right)\left\\{\overleftarrow{\partial}_{\zeta}\frac{\mathbb{T}(\zeta)}{\sqrt{f}}+\mathbb{\Gamma}\left(\beta
k^{0}+\frac{1}{2}\partial_{\zeta}\ln
f\right)-\frac{\beta}{2}\Gamma^{(i)}k_{i}\right\\}=0\ ,$ (95)

and

$\left\\{\frac{\mathbb{T}(\zeta)}{\sqrt{f}}\partial_{\zeta}+\mathbb{\Gamma}\left(\beta
k^{0}+\frac{1}{2}\partial_{\zeta}\ln
f\right)-\frac{\beta}{2}\Gamma^{(i)}k_{i}\right\\}\left(e^{-\beta
k^{0}\zeta}\mathbb{T}(\zeta)r^{d/2}\Psi_{1}(\zeta,-k)\right)=0\ ,$ (96)

respectively. Note that if $\Psi_{1}(\zeta,k)$ is a solution of the Dirac
equation, then so is its time reversal $e^{-\beta
k^{0}\zeta}\mathbb{T}(\zeta)r^{d/2}\Psi_{1}(\zeta,-k)$, which is what we have
used in the second equation above. Multiplying Eq. (95) from the right by
$\sqrt{f}e^{-\beta k^{0}\zeta}r^{d/2}\Psi_{1}(\zeta,-k)$, Eq. (96) from the
left by $\sqrt{f}r^{d/2}\overline{\Psi}_{2}(\zeta,k)\mathbb{T}(\zeta)$ and
adding, we obtain999Here, we have used the bulk gamma matrix identities
$\Gamma^{(i)}+\mathbb{T}(\zeta)\Gamma^{(i)}\mathbb{T}(\zeta)=0\
,\quad\text{and}\quad\sqrt{f}\left[\mathbb{\Gamma}+\mathbb{T}(\zeta)\mathbb{\Gamma}\mathbb{T}(\zeta)\right]=\mathbb{T}(\zeta)\
.$ (97)

$\begin{split}&\partial_{\zeta}\left\\{r^{d/2}\overline{\Psi}_{2}(\zeta,k)\mathbb{T}(\zeta)e^{-\beta
k^{0}\zeta}r^{d/2}\Psi_{1}(\zeta,-k)\right\\}\\\ &\hskip 56.9055pt+\left(\beta
k^{0}+\frac{1}{2}\partial_{\zeta}\ln
f\right)\left\\{r^{d/2}\overline{\Psi}_{2}(\zeta,k)\mathbb{T}(\zeta)e^{-\beta
k^{0}\zeta}r^{d/2}\Psi_{1}(\zeta,-k)\right\\}=0\ ,\end{split}$ (98)

which integrates to

$\partial_{\zeta}\mathfrak{W}(\zeta,k)=0\
,\quad\text{where}\quad\mathfrak{W}(\zeta,k)\equiv\sqrt{f}r^{d/2}\overline{\Psi}_{2}(\zeta,k)\mathbb{T}(\zeta)r^{d/2}\Psi_{1}(\zeta,-k)\
.$ (99)

Note that this conserved quantity contains the boundary sources. To construct
a bulk-to-bulk propagator we would require a quantity independent of boundary
sources. So, we now define a square matrix $\mathbb{W}(\zeta,k)$ in the
boundary spinor space where we strip off the boundary sources from this
conserved quantity $\mathfrak{W}(\zeta,k)$, given by

$\mathbb{W}[\bar{s}_{2}(\zeta,k),s_{1}(\zeta,k)]\equiv\sqrt{f}r^{d/2}\bar{s}_{2}(\zeta,-k)\mathbb{T}(\zeta)r^{d/2}s_{1}(\zeta,k)\
.$ (100)

Here, $\bar{s}_{2}(\zeta,k)$ and $s_{1}(\zeta,k)$ are the conjugate boundary-
to-bulk and boundary-to-bulk propagators respectively.

This boundary spinor-space matrix in the basis of ingoing and outgoing
solutions takes the form

$\begin{split}\lim_{\zeta\to
0,1}\mathbb{W}[\overline{S}^{\text{in}}(\zeta,k),S^{\text{in}}(\zeta,k)]&=\lim_{\zeta\to
0,1}\sqrt{f}r^{d/2}\overline{S}^{\text{in}}(\zeta,-k)\mathbb{T}(\zeta)r^{d/2}S^{\text{in}}(\zeta,k)\\\
&=\lim_{\zeta\to
0,1}r^{d/2}\overline{S}^{\text{in}}(\zeta,-k)(\mathbb{P}_{{}_{+}}-\mathbb{P}_{{}_{-}})r^{d/2}S^{\text{in}}(\zeta,k)\\\
&=-\left\\{\mathfrak{K}_{\rm ret}(k)+\mathfrak{K}_{\rm ret}(-k)\right\\}\
,\end{split}$ (101)

where we have used Eq. (31), Eq. (83) and have also defined the boundary
retarded two-point correlator $\mathfrak{K}_{\rm ret}(k)$ as

$\mathfrak{K}_{\rm ret}(k)\equiv\lim_{\zeta\to
0,1}\overline{S}_{0}\mathbb{P}_{{}_{-}}r^{d/2}S^{\text{in}}(\zeta,k)=-\lim_{\zeta\to
0,1}r^{d/2}\overline{S}^{\text{in}}(\zeta,k)\mathbb{P}_{{}_{+}}S_{0}\ .$ (102)

By similar computations, we find the other combinations:

$\begin{split}&\mathbb{W}[\overline{S}^{\text{in}}(\zeta,k),S^{\text{in}}(\zeta,k)]=-\left\\{\mathfrak{K}_{\rm
ret}(k)+\mathfrak{K}_{\rm
ret}(-k)\right\\}=-\mathbb{W}[\overline{S}^{\text{out}}(\zeta,k),S^{\text{out}}(\zeta,k)]\
,\\\
&\mathbb{W}[\overline{S}^{\text{out}}(\zeta,k),S^{\text{in}}(\zeta,k)]=0=\mathbb{W}[\overline{S}^{\text{in}}(\zeta,k),S^{\text{out}}(\zeta,k)]\
.\end{split}$ (103)

Similar expressions for the conserved quantity corresponding to the right and
the left boundary-to-bulk propagators can be derived from the above equations
using Eq. (36) and Eq. (88). Explicitly, we have

$\begin{split}&\mathbb{W}[\bar{s}_{{}_{\text{R}}}(\zeta,k),s_{{}_{\text{R}}}(\zeta,k)]=0=\mathbb{W}[\bar{s}_{{}_{\text{L}}}(\zeta,k),s_{{}_{\text{L}}}(\zeta,k)]\
,\\\
&\mathbb{W}[\bar{s}_{{}_{\text{R}}}(\zeta,k),s_{{}_{\text{L}}}(\zeta,k)]=n^{{}_{\text{FD}}}_{k^{0}}\left\\{\mathfrak{K}_{\rm
ret}(k)+\mathfrak{K}_{\rm ret}(-k)\right\\}=e^{-\beta
k^{0}}\mathbb{W}[\bar{s}_{{}_{\text{L}}}(\zeta,k),s_{{}_{\text{R}}}(\zeta,k)]\
.\end{split}$ (104)

#### Determining the spinor bulk-to-bulk propagator

Let us now take a look at the differential equation satisfied by the spinorial
bulk-to-bulk propagator (Eq. (43)). Away from the point $\zeta=\zeta_{0}$, it
is simply the free Dirac equation and therefore in these regions, the bulk-to-
bulk propagator has to be proportional to the boundary-to-bulk propagators.
Keeping in mind the boundary conditions in Eq. (44), we write an ansatz for
the bulk-to-bulk propagator as

$\begin{split}\mathbb{S}(\zeta|\zeta_{0},k)&=\left[{s}_{{}_{\text{R}}}(\zeta,k)M_{<}(\zeta_{0},k)\bar{s}_{{}_{\text{L}}}(\zeta_{0},-k)\right]\Theta_{{}^{\text{SK}}}(\zeta<\zeta_{0})\\\
&\hskip
1.42271pt+\left[{s}_{{}_{\text{L}}}(\zeta,k)M_{>}(\zeta_{0},k)\bar{s}_{{}_{\text{R}}}(\zeta_{0},-k)\right]\Theta_{{}^{\text{SK}}}(\zeta>\zeta_{0})\
.\end{split}$ (105)

Here, the symbol $\Theta_{{}^{\text{SK}}}$ denotes a Heaviside theta function
on the grSK contour defined as

$\begin{split}&\Theta_{{}^{\text{SK}}}(\zeta<\zeta_{0})\equiv\Bigg{\\{}\begin{array}[]{cc}\zeta&\text{if
$\zeta$ comes before $\zeta_{0}$ on the grSK contour}\\\ \zeta_{0}&\text{if
$\zeta_{0}$ comes before $\zeta$ on the grSK contour}\end{array}\ ,\\\
&\Theta_{{}^{\text{SK}}}(\zeta>\zeta_{0})\equiv
1-\Theta_{{}^{\text{SK}}}(\zeta<\zeta_{0})\ .\end{split}$ (106)

The symbols $M_{<}(\zeta_{0},k)$ and $M_{>}(\zeta_{0},k)$ denote square
matrices in the boundary spinor space which we will soon determine using the
jump conditions at $\zeta=\zeta_{0}$. It is easy to see that the above ansatz
is a square matrix in the bulk spinor space and also that it satisfies the
binormalisable boundary conditions given in Eq. (44).

The jump condition is obtained by integrating the delta-sourced Dirac equation
in Eq.(43) around the point $\zeta=\zeta_{0}$. Explicitly, we have

$\begin{split}\frac{i\beta
r}{2}r^{d/2}\Gamma^{\scalebox{0.9}{$\scriptscriptstyle
A$}}D_{\scalebox{0.9}{$\scriptscriptstyle
A$}}\mathbb{S}(\zeta|\zeta_{0},k)&=\left\\{\frac{\mathbb{T}(\zeta)}{\sqrt{f}}\partial_{\zeta}+\mathbb{\Gamma}\left(\beta
k^{0}+\frac{1}{2}\partial_{\zeta}\ln
f\right)-\frac{\beta}{2}\Gamma^{(i)}k_{i}\right\\}\left(r^{d/2}\mathbb{S}(\zeta|\zeta_{0},k)\right)\\\
&=\mathbb{1}\frac{\delta(\zeta-\zeta_{0})}{r^{d/2}f}\ ,\end{split}$ (107)

where we have used the fact that $\sqrt{-g}=\frac{i\beta}{2}r^{d+1}f$.
Integrating the above ODE from $\zeta=\zeta_{0}^{-}$ to $\zeta=\zeta_{0}^{+}$,
we arrive at the jump condition

$\frac{\mathbb{T}(\zeta)}{\sqrt{f}}r^{d/2}\mathbb{S}(\zeta|\zeta_{0},k)\Bigg{|}_{\zeta=\zeta_{0}^{-}}^{\zeta=\zeta_{0}^{+}}=\frac{\mathbb{1}}{r_{0}^{d/2}f(\zeta_{0})}\
.$ (108)

Using the ansatz in Eq. (105), the above jump condition takes the form

$\sqrt{f}r^{d}\mathbb{T}(\zeta)\left\\{{s}_{{}_{\text{L}}}(\zeta,k)\
M_{>}(\zeta,k)\
\bar{s}_{{}_{\text{R}}}(\zeta,-k)-{s}_{{}_{\text{R}}}(\zeta,k)\
M_{<}(\zeta,k)\ \bar{s}_{{}_{\text{L}}}(\zeta,-k)\right\\}=\mathbb{1}\ .$
(109)

Notice now that pre-multiplying and post-multiplying the above equation with
appropriate boundary-to-bulk propagators, we obtain simple equations involving
the conserved quantity that we computed in Eq. (100) and Eq. (104). Pre-
multiplying by $r^{d/2}\bar{s}_{{}_{\text{L}}}(\zeta,-k)$ and post-multiplying
by $\sqrt{f}\mathbb{T}r^{d/2}s_{{}_{\text{R}}}(\zeta,k)$, we obtain
$M_{<}(\zeta,k)$. Similarly, pre-multiplying by
$r^{d/2}\bar{s}_{{}_{\text{R}}}(\zeta,-k)$ and post-multiplying by
$\sqrt{f}\mathbb{T}r^{d/2}s_{{}_{\text{L}}}(\zeta,k)$, we obtain
$M_{>}(\zeta,k)$. The explicit expressions are as follows.

$M_{<}(\zeta,k)=-\frac{1}{(1-n^{{}_{\text{FD}}}_{k^{0}})\left\\{\mathfrak{K}_{\rm
ret}(k)+\mathfrak{K}_{\rm ret}(-k)\right\\}}\ ,\quad
M_{>}(\zeta,k)=\frac{1}{n^{{}_{\text{FD}}}_{k^{0}}\left\\{\mathfrak{K}_{\rm
ret}(k)+\mathfrak{K}_{\rm ret}(-k)\right\\}}\ .$ (110)

Substituting these results in the ansatz in Eq. (105), we obtain

$\begin{split}\mathbb{S}(\zeta|\zeta_{0},k)&=-\left[{s}_{{}_{\text{R}}}(\zeta,k)\frac{1}{(1-n^{{}_{\text{FD}}}_{k^{0}})\left\\{\mathfrak{K}_{\rm
ret}(k)+\mathfrak{K}_{\rm
ret}(-k)\right\\}}\bar{s}_{{}_{\text{L}}}(\zeta_{0},-k)\right]\Theta_{{}^{\text{SK}}}(\zeta<\zeta_{0})\\\
&\quad+\left[{s}_{{}_{\text{L}}}(\zeta,k)\frac{1}{n^{{}_{\text{FD}}}_{k^{0}}\left\\{\mathfrak{K}_{\rm
ret}(k)+\mathfrak{K}_{\rm
ret}(-k)\right\\}}\bar{s}_{{}_{\text{R}}}(\zeta_{0},-k)\right]\Theta_{{}^{\text{SK}}}(\zeta>\zeta_{0})\
.\end{split}$ (111)

We take a moment to also note that the $\mathbb{P}_{-}$ projected boundary
limits of the above propagator take the form

$\lim_{\zeta\to
0}r^{\frac{d}{2}}\overline{S}_{0}\mathbb{P}_{-}\mathbb{S}(\zeta|\zeta_{0},k)=-\overline{s}_{{}_{\text{L}}}(\zeta_{0},k)\
,\qquad\lim_{\zeta\to
1}r^{\frac{d}{2}}\overline{S}_{0}\mathbb{P}_{-}\mathbb{S}(\zeta|\zeta_{0},k)=-\overline{s}_{{}_{\text{R}}}(\zeta_{0},k)\
.$ (112)

### Retarded and advanced spinor bulk-to-bulk propagators

Now that we have the binormalisable bulk-to-bulk propagator, we can also
construct bulk-to-bulk propagators that have specific causal properties, i.e.,
the retarded and the advanced bulk-to-bulk propagators. These propagators are
defined to be normalisable at the left boundary, and analytic in the upper-
half and the lower-half of the frequency plane respectively. Here, we will not
provide a derivation of these propagators, as it is similar to the computation
of the binormalisable bulk-to-bulk propagator given above. We just quote the
results:

$\begin{split}\mathbb{S}_{\text{ret}}(\zeta|\zeta_{0},k)&=\left[{s}_{{}_{\text{R}}}(\zeta,k)\frac{1}{(1-n^{{}_{\text{FD}}}_{k^{0}})\left\\{\mathfrak{K}_{\rm
ret}(k)+\mathfrak{K}_{\rm
ret}(-k)\right\\}}\overline{S}^{\text{out}}(\zeta_{0},-k)\right]\Theta_{{}^{\text{SK}}}(\zeta<\zeta_{0})\\\
&\quad-\left[S^{\text{in}}(\zeta,k)\frac{1}{n^{{}_{\text{FD}}}_{k^{0}}\left\\{\mathfrak{K}_{\rm
ret}(k)+\mathfrak{K}_{\rm
ret}(-k)\right\\}}\bar{s}_{{}_{\text{R}}}(\zeta_{0},-k)\right]\Theta_{{}^{\text{SK}}}(\zeta>\zeta_{0})\
,\end{split}$ (113)

and

$\begin{split}\mathbb{S}_{\text{adv}}(\zeta|\zeta_{0},k)&=\left[{s}_{{}_{\text{R}}}(\zeta,k)\frac{1}{(1-n^{{}_{\text{FD}}}_{k^{0}})\left\\{\mathfrak{K}_{\rm
ret}(k)+\mathfrak{K}_{\rm
ret}(-k)\right\\}}\overline{S}^{\text{in}}(\zeta_{0},-k)\right]\Theta_{{}^{\text{SK}}}(\zeta<\zeta_{0})\\\
&\quad-\left[S^{\text{out}}(\zeta,k)\frac{1}{n^{{}_{\text{FD}}}_{k^{0}}\left\\{\mathfrak{K}_{\rm
ret}(k)+\mathfrak{K}_{\rm
ret}(-k)\right\\}}\bar{s}_{{}_{\text{R}}}(\zeta_{0},-k)\right]\Theta_{{}^{\text{SK}}}(\zeta>\zeta_{0})\
.\end{split}$ (114)

Now, we show how these propagators satisfy all the required properties. First
of all, we note that the right boundary-to-bulk propagator
$s_{{}_{\text{R}}}(\zeta,k)$ satisfies

$\lim_{\zeta\rightarrow
0}r^{\frac{d}{2}}\mathbb{P}_{+}{s}_{{}_{\text{R}}}(\zeta,k)=0\ .$ (115)

This implies that the retarded and the advanced bulk-to-bulk propagators
satisfy

$\lim_{\zeta\rightarrow
0}r^{\frac{d}{2}}\mathbb{P}_{+}\mathbb{S}_{\text{ret}}(\zeta|\zeta_{0},k)=0\
,\qquad\lim_{\zeta\rightarrow
0}r^{\frac{d}{2}}\mathbb{P}_{+}\mathbb{S}_{\text{adv}}(\zeta|\zeta_{0},k)=0\
.$ (116)

Next, we have to show that the above propagators have the desired analyticity
properties on the complex frequency plane. To do this, we will employ a
similar set of arguments as the one used in the case of the scalar bulk-to-
bulk propagator. To begin with, we use the intuition that the ingoing
boundary-to-bulk propagator $S^{\text{in}}(\zeta,k)$ has the same pole
structure, i.e., the quasinormal poles, on the complex frequency plane as the
boundary retarded correlator $\mathfrak{K}_{\rm ret}(k)$. We can then use the
same trick we used for the case of the scalar retarded and advanced
propagators to conclude that $\mathbb{S}_{\text{ret}}(\zeta|\zeta_{0},k)$, has
the same poles as $\mathfrak{K}_{\rm ret}(k)$ and is analytic in the upper
half of the frequency plane. Similarly,
$\mathbb{S}_{\text{adv}}(\zeta|\zeta_{0},k)$ is analytic in the lower half of
the frequency plane.

Here, we point out that the binormalisable spinor bulk-to-bulk propagator can
be written as a linear combination of the retarded and the advanced bulk-to-
bulk propagators as

$\mathbb{S}(\zeta|\zeta_{0},k)=\
n^{{}_{\text{FD}}}_{k^{0}}\mathbb{S}_{\text{ret}}(\zeta|\zeta_{0},k)+(1-n^{{}_{\text{FD}}}_{k^{0}})\mathbb{S}_{\text{adv}}(\zeta|\zeta_{0},k)\
.$ (117)

This relation can be easily obtained by replacing the corresponding
propagators, and $n^{{}_{\text{BE}}}_{k^{0}}$ with
$-n^{{}_{\text{FD}}}_{k^{0}}$, from the corresponding relation for scalar
bulk-to-bulk propagators GodwinLoga :

$\mathbb{G}(\zeta|\zeta_{0},k)=\
-n^{{}_{\text{BE}}}_{k^{0}}\mathbb{G}_{\text{ret}}(\zeta|\zeta_{0},k)+(1+n^{{}_{\text{BE}}}_{k^{0}})\mathbb{G}_{\text{adv}}(\zeta|\zeta_{0},k)\
.$ (118)

##  Appendix C Monodromy integrals around the grSK contour

The computation of the four-point exchange diagrams that we have detailed in
the main text requires performing integrals of the form

$\mathscr{I}_{\mathbb{G}}\equiv\oint_{\zeta_{1}}\oint_{\zeta_{2}}\
[f(\zeta_{1})]^{\frac{\alpha_{1}}{2}}[f(\zeta_{2})]^{\frac{\alpha_{2}}{2}}e^{-\beta
k^{0}_{1}\zeta_{1}}e^{-\beta
k^{0}_{2}\zeta_{2}}\mathbb{G}(\zeta_{2}|\zeta_{1},p)\mathscr{F}(\zeta_{1},\zeta_{2})\
,$ (119)

and

$\mathscr{I}_{\mathbb{S}}\equiv\oint_{\zeta_{1}}\oint_{\zeta_{2}}\
[f(\zeta_{1})]^{\frac{\alpha_{1}}{2}}[f(\zeta_{2})]^{\frac{\alpha_{2}}{2}}e^{-\beta
k^{0}_{1}\zeta_{1}}e^{-\beta
k^{0}_{2}\zeta_{2}}\bar{\mathfrak{a}}_{2}(\zeta_{2})\mathbb{S}(\zeta_{2}|\zeta_{1},p)\mathfrak{a}_{1}(\zeta_{1})\
.$ (120)

Here $\alpha_{1}$ and $\alpha_{2}$ are integers,
$\mathscr{F}(\zeta_{1},\zeta_{2})$ is a scalar analytic function on the
complex $r_{1}$ and $r_{2}$ planes, and $\mathfrak{a}_{1}(\zeta)$ and
$\bar{\mathfrak{a}}_{2}(\zeta)$ are analytic functions on the $r$ plane that
has the spinor structure of a spinor and a conjugate spinor boundary-to-bulk
propagator respectively.

The double-discontinuity integrals above reduce to integrals purely on one
copy of the black hole exterior. That is, the monodromy around the grSK branch
cut can be explicitly computed and we can write the answer for the contour
integral as an integral over the real $r$ line. In this appendix, we will show
this in detail for the second integral above, i.e., the one involving the
spinor bulk-to-bulk propagator, $\mathscr{I}_{\mathbb{S}}$.

On the other hand, the double discontinuity integral of the scalar bulk-to-
bulk propagator has been computed in GodwinLoga . Here the authors were
concerned only with scalar interactions and thus do not include the factors of
$\sqrt{f}$ we have above. Nevertheless, including the new factors of
$\sqrt{f}$ is relatively straightforward and we just present the result here.

$\begin{split}\mathscr{I}_{\mathbb{G}}=\int_{\rm ext_{1}}\int_{\rm
ext_{2}}[f(\zeta_{1})]^{\frac{\alpha_{1}}{2}}[f(\zeta_{2})]^{\frac{\alpha_{2}}{2}}e^{-\beta
k^{0}_{1}\zeta_{1}}e^{-\beta k^{0}_{2}\zeta_{2}}\mathbb{G}_{\rm
DD}(\zeta_{2}|\zeta_{1},p)\mathscr{F}(\zeta_{1},\zeta_{2})\ ,\end{split}$
(121)

where $\int_{\rm ext}$ denotes the integral over the exterior of the left copy
of the black bole, defined in Eq. (54). The symnbol $\mathbb{G}_{\rm
DD}(\zeta_{2}|\zeta_{1},p)$ denotes the double-discontinuity given by

$\begin{split}\mathbb{G}_{\rm
DD}(\zeta_{2}|\zeta_{1},p)&\equiv\frac{-n^{{}_{\text{BE}}}_{p^{0}}}{\left(1+\frac{1}{(-)^{\alpha_{1}}e^{\beta(k_{1}^{0}-p^{0})}-1}\right)\left(1+\frac{1}{(-)^{\alpha_{2}}e^{\beta
k_{2}^{0}}-1}\right)}\mathbb{G}_{\text{ret}}(\zeta_{1}|\zeta_{1},p)\\\
&+\frac{1+n^{{}_{\text{BE}}}_{p^{0}}}{\left(1+\frac{1}{(-)^{\alpha_{1}}e^{\beta
k_{1}^{0}}-1}\right)\left(1+\frac{1}{(-)^{\alpha_{2}}e^{\beta(k_{2}^{0}+p^{0})}-1}\right)}\mathbb{G}_{\text{adv}}(\zeta_{1}|\zeta_{1},p)\
.\end{split}$ (122)

Here $\mathbb{G}_{\text{ret}}(\zeta_{1}|\zeta_{1},p)$ and
$\mathbb{G}_{\text{adv}}(\zeta_{1}|\zeta_{1},p)$ are the retarded and the
advanced bulk-to-bulk propagators respectively, given in Eq. (92). From the
above result, note that the temperature-dependent factors outside the
propagators are all either Fermi-Dirac or Bose-Einstein statistical factors,
depending on the parity of $\alpha_{1}$ and $\alpha_{2}$. In particular, note
that for $\alpha_{1}=\alpha_{2}=0$, we obtain only Bose-Einstein factors and
that the result matches the purely scalar theory result in GodwinLoga .

We now come to the double-discontinuity involving the spinor bulk-to-bulk
propagator. Before proceeding to perform the monodromy integrals, we take a
moment to note that exactly as in the purely scalar case, here too there are
no pole contributions from the horizon. This might seem surprising in the case
of fermions since there are factors of $1/\sqrt{f}$ that can multiply to give
horizon poles of the form $1/f$. This does not happen because the factors of
$1/\sqrt{f}$ in all our computations only come from the outgoing propagator.
In particular, from the time-reversal matrix. Since the time-reversal matrix
is idempotent (squares to identity), there can never be factors of $1/f$ in
the calculation, thus avoiding the question of horizon-pole contributions.

Accounting for the monodromy across the branch cut, the double-discontinuity
of the spinor bulk-to-bulk propagator takes the form

$\begin{split}\mathscr{I}_{\mathbb{S}}=\int_{\text{ext}_{1}}\int_{\text{ext}_{2}}\
[f(\zeta_{1})]^{\frac{\alpha_{1}}{2}}[f(\zeta_{2})]^{\frac{\alpha_{2}}{2}}e^{-\beta
k^{0}_{1}\zeta_{1}}e^{-\beta
k^{0}_{2}\zeta_{2}}\bar{\mathfrak{a}}_{2}(\zeta_{2})\mathbb{S}_{{}_{\text{DD}}}(\zeta_{2}|\zeta_{1},p)\bar{\mathfrak{a}}_{1}(\zeta_{1})\
,\end{split}$ (123)

where $\mathbb{S}_{{}_{\text{DD}}}(\zeta_{2}|\zeta_{1},p)$ is the double
discontinuity

$\begin{split}\mathbb{S}_{{}_{\text{DD}}}(\zeta_{2}|\zeta_{1},p)&\equiv\mathbb{S}(\zeta_{2}|\zeta_{1},p)-(-)^{\alpha_{1}}e^{-\beta
k^{0}_{1}}\mathbb{S}(\zeta_{2}|\zeta_{1}+1,p)\\\ &-(-)^{\alpha_{2}}e^{-\beta
k^{0}_{2}}\mathbb{S}(\zeta_{2}+1|\zeta_{1},p)+(-)^{\alpha_{1}+\alpha_{2}}e^{-\beta
k^{0}_{1}}e^{-\beta k^{0}_{2}}\mathbb{S}(\zeta_{2}+1|\zeta_{1}+1,p)\
.\end{split}$ (124)

This expression can now be greatly simplified using the theta function
identities

$\begin{split}&\Theta_{{}^{\text{SK}}}(\zeta_{1}+1>\zeta_{2}+1)=\Theta_{{}^{\text{SK}}}(\zeta_{1}>\zeta_{2})\
,\\\ &\Theta_{{}^{\text{SK}}}(\zeta_{1}+1>\zeta_{2})=1\
,\qquad\Theta_{{}^{\text{SK}}}(\zeta_{1}>\zeta_{2}+1)=0\ .\end{split}$ (125)

These hold when $\zeta_{1}$ and $\zeta_{2}$ both belong to the upper branch of
the grSK contour, which is indeed the case here. With the explicit form of the
bulk-to-bulk propagator as well, the double-discontinuity can be written as

$\begin{split}\mathbb{S}_{{}_{\text{DD}}}(\zeta_{2}|\zeta_{1},p)&=S^{\text{in}}(\zeta_{2},p)\frac{1}{\left\\{\mathfrak{K}_{\rm
ret}(p)+\mathfrak{K}_{\rm
ret}(-p)\right\\}}\left[c_{1}\overline{S}^{\text{in}}(\zeta_{1},-p)-c_{2}\overline{S}^{\text{out}}(\zeta_{1},-p)\right]\\\
&+S^{\text{out}}(\zeta_{2},p)\frac{1}{\left\\{\mathfrak{K}_{\rm
ret}(p)+\mathfrak{K}_{\rm
ret}(-p)\right\\}}\left[c_{3}\overline{S}^{\text{in}}(\zeta_{1},-p)-c_{4}\overline{S}^{\text{out}}(\zeta_{1},-p)\right]\
,\end{split}$ (126)

where

$\begin{split}c_{2}&=n^{{}_{\text{FD}}}_{p^{0}}\left(e^{i\alpha_{1}\pi}e^{-\beta(k_{1}^{0}-p^{0})}+1\right)\left(e^{i\alpha_{2}\pi}e^{-\beta
k_{2}^{0}}-1\right)\ ,\\\
c_{3}&=(1-n^{{}_{\text{FD}}}_{p^{0}})\left(e^{i\alpha_{1}\pi}e^{-\beta
k_{1}^{0}}-1\right)\left(e^{i\alpha_{2}\pi}e^{-\beta(k_{2}^{0}+p^{0})}+1\right)\
,\\\
c_{1}&=(1+c_{2})\Theta_{{}^{\text{SK}}}(\zeta_{2}<\zeta_{1})-(1+c_{3})\Theta_{{}^{\text{SK}}}(\zeta_{2}>\zeta_{1})\
,\\\
c_{4}&=(1+c_{3})\Theta_{{}^{\text{SK}}}(\zeta_{2}<\zeta_{1})-(1+c_{2})\Theta_{{}^{\text{SK}}}(\zeta_{2}>\zeta_{1})\
.\end{split}$ (127)

Notice that the factors in $c_{2}$ and $c_{3}$ are just Fermi-Dirac or Bose-
Einstein statistical factors depending on the parity of $\alpha_{1}$ and
$\alpha_{2}$. Moreover, there are only two independent combinations of
statistical factors here. Thus we can indeed rewrite the above expression as

$\begin{split}\mathbb{S}_{{}_{\text{DD}}}(\zeta_{2}|\zeta_{1},p)&=-\left(e^{i\alpha_{1}\pi}e^{-\beta(k^{0}_{1}-p^{0})}+1\right)\left(e^{i\alpha_{2}\pi}e^{-\beta
k^{0}_{2}}-1\right)n^{{}_{\text{FD}}}_{p^{0}}\mathbb{S}_{\text{ret}}(\zeta_{2}|\zeta_{1},p)\\\
&-\left(e^{i\alpha_{1}\pi}e^{-\beta
k^{0}_{1}}-1\right)\left(e^{i\alpha_{2}\pi}e^{-\beta(k^{0}_{2}+p^{0})}+1\right)(1-n^{{}_{\text{FD}}}_{p^{0}})\mathbb{S}_{\text{adv}}(\zeta_{2}|\zeta_{1},p)\
,\end{split}$ (128)

where $\mathbb{S}_{\text{ret}}(\zeta_{2}|\zeta_{1},p)$ and
$\mathbb{S}_{\text{adv}}(\zeta_{2}|\zeta_{1},p)$ are the retarded and the
advanced bulk-to-bulk propagators given in Eq. (113) and Eq. (114)
respectively.

So far, we have focused on evaluating double-discontinuity integrals relevant
to the computation of the four-point exchange influence phase. To evaluate
higher point functions, we require higher monodromy integrals with richer
structures of products of bulk-to-bulk propagators. For example, at the level
of five-point functions, we require integrals of the form

$\oint_{\zeta_{1}}\oint_{\zeta_{2}}\oint_{\zeta_{3}}\
[f(\zeta_{1})]^{\frac{\alpha_{1}}{2}}[f(\zeta_{2})]^{\frac{\alpha_{2}}{2}}[f(\zeta_{3})]^{\frac{\alpha_{3}}{2}}e^{-\beta(k^{0}_{1}\zeta_{1}+k^{0}_{2}\zeta_{2}+k^{0}_{3}\zeta_{3})}\mathbb{G}(\zeta_{2}|\zeta_{1},p_{1})\bar{\mathfrak{a}}_{2}(\zeta_{3},\zeta_{1})\mathbb{S}(\zeta_{3}|\zeta_{2},p_{2})\mathfrak{a}_{1}(\zeta_{2})$
(129)

and

$\oint_{\zeta_{1}}\oint_{\zeta_{2}}\oint_{\zeta_{3}}\
[f(\zeta_{1})]^{\frac{\alpha_{1}}{2}}[f(\zeta_{2})]^{\frac{\alpha_{2}}{2}}[f(\zeta_{3})]^{\frac{\alpha_{3}}{2}}e^{-\beta(k^{0}_{1}\zeta_{1}+k^{0}_{2}\zeta_{2}+k^{0}_{3}\zeta_{3})}\bar{\mathfrak{a}}_{2}(\zeta_{3},\zeta_{2})\mathbb{S}(\zeta_{3}|\zeta_{2},p_{2})\mathbb{S}(\zeta_{2}|\zeta_{1},p_{1})\mathfrak{a}_{1}(\zeta_{1})\
.$ (130)

These integrals can also be computed by the techniques presented above and can
also be arranged in terms of the retarded and the advanced propagators for
scalars and spinors, presented above. Since we are only interested in the
four-point functions, we do not present these results here.

##  Appendix D Formal perturbative expansion of the influence phase

In this appendix, we will present a formal perturbative expansion of the
Yukawa influence phase in terms of the free solutions and the bulk-to-bulk
propagators.

The Euler-Lagrange equations obtained by varying the Yukawa action in Eq. (37)
are

$\Box\phi=i\lambda\bar{\Psi}\
\qquad\text{and}\qquad\Psi\Gamma^{\scalebox{0.9}{$\scriptscriptstyle
A$}}D_{\scalebox{0.9}{$\scriptscriptstyle A$}}\Psi=\lambda\phi\Psi\ ,$ (131)

where $\Box$ is the wave operator of the AdSd+1 Schwarzschild blackbrane, and
$D_{\scalebox{0.9}{$\scriptscriptstyle A$}}$ is the covariant derivative
acting on the spinor fields defined in Eq. (26). These equations can now be
solved perturbatively in the Yukawa coupling $\lambda$. In other words, we
will write the solutions in the form given in Eq. (39) and Eq. (40). The
leading-order solutions $\Psi_{(0)}$ and $\phi_{(0)}$ solve the free Dirac
equation and the free wave equation respectively. The higher-order solutions
can be found by solving the equations of motion above perturbatively.

We take the leading-order solutions to satisfy the GKPW boundary conditions,
and thus the higher-order solutions must all be normalisable on both the
boundaries. Therefore, we can write expressions of the form

$\Psi_{(n)}(\zeta,k)=\oint_{\zeta}\
\mathbb{S}(\zeta|\zeta_{0},k)\mathfrak{J}_{(n)}(\zeta_{0},k)\
,\quad\text{and}\quad\phi_{(n)}(\zeta,k)=\oint_{\zeta}\
\mathbb{G}(\zeta|\zeta_{0},k)\mathbb{J}_{(n)}(\zeta_{0},k)\ ,$ (132)

where $\mathbb{S}(\zeta|\zeta_{0},k)$ and $\mathbb{G}(\zeta|\zeta_{0},k)$ are
the spinor and scalar bulk-to-propagators, which are normalisable on both the
boundaries, respectively. Here $\mathfrak{J}_{(n)}(\zeta_{0},k)$ and
$\mathbb{J}_{(n)}(\zeta_{0},k)$ are sources for the $n$th order solutions,
which can be read-off from the equations of motion. For example, at the first
and the second orders, we have the expressions given in Eq. (46) and Eq. (47)
respectively.

The above perturbative expansions of the solutions can now be substituted into
the action to provide the on-shell action, which is the influence phase at
leading order.101010Here, leading order refers to the leading term in $1/N$
expansion in the boundary theory. We will arrange this expansion with the
notation given in Eq. (48). The symbol $S_{(2)}$ denotes the part of the on-
shell action that is quadratic in the boundary sources and takes the form

$S_{(2)}=-\frac{1}{2}\int\mathop{}\\!\mathrm{d}\Sigma^{\scalebox{0.9}{$\scriptscriptstyle
A$}}\phi_{(0)}\partial_{\scalebox{0.9}{$\scriptscriptstyle
A$}}\phi_{(0)}+\int\mathop{}\\!\mathrm{d}^{d}\mathbf{x}\
r^{d}\left(i\overline{\Psi}_{(0)}\mathbb{P}_{-}\Psi_{(0)}\right)\Big{|}_{\zeta=0}^{\zeta=1}\
.$ (133)

The term linear in the interaction parameter $\lambda$ is also cubic in the
boundary sources and takes the form

$\begin{split}S_{(3)}&=\oint\mathop{}\\!\mathrm{d}\zeta\int\mathop{}\\!\mathrm{d}^{d}\mathbf{x}\sqrt{-g}\
\overline{\Psi}_{(0)}i\Gamma^{\scalebox{0.9}{$\scriptscriptstyle
A$}}D_{\scalebox{0.9}{$\scriptscriptstyle
A$}}\Psi_{(1)}-\oint\mathop{}\\!\mathrm{d}\zeta\int\mathop{}\\!\mathrm{d}^{d}\mathbf{x}\sqrt{-g}\
\partial_{\scalebox{0.9}{$\scriptscriptstyle
A$}}\phi_{(0)}\partial^{\scalebox{0.9}{$\scriptscriptstyle A$}}\phi_{(1)}\\\
&+\int\mathop{}\\!\mathrm{d}^{d}\mathbf{x}\
r^{d}\left[i\overline{\Psi}_{(0)}\mathbb{P}_{-}\Psi_{(1)}+i\overline{\Psi}_{(1)}\mathbb{P}_{-}\Psi_{(0)}\right]\Big{|}_{\zeta=0}^{\zeta=1}-i\oint\mathop{}\\!\mathrm{d}\zeta\int\mathop{}\\!\mathrm{d}^{d}\mathbf{x}\sqrt{-g}\
\phi_{(0)}\overline{\Psi}_{(0)}\Psi_{(0)}\ .\end{split}$ (134)

This expression can be greatly simplified using the equations of motion
satisfied by the fields. Firstly, note that the first-order corrections (in
the solutions) satisfy Eq. (45) with $n=1$. Using the explicit form of the
$n=1$ sources in Eq. (46), we see that the first term cancels the final term.
The second term vanishes, as can be seen by using integration by parts and
noting that the first-order solution vanishes at the boundary. This leaves
only the boundary term in the square brackets. Again, the second term here
vanishes because of the boundary conditions on $\overline{\Psi}_{(1)}$. Thus,
we finally get only one simple term

$\begin{split}S_{(3)}&=\int\mathop{}\\!\mathrm{d}^{d}\mathbf{x}\ r^{d}\
i\overline{\Psi}_{(0)}\mathbb{P}_{-}\Psi_{(1)}\Big{|}_{\zeta=0}^{\zeta=1}=\
ir^{d}\int\frac{\mathop{}\\!\mathrm{d}^{d}k}{(2\pi)^{d}}\overline{\Psi}_{(0)}(-k)\mathbb{P}_{{}_{-}}\Psi_{(1)}(k)\Big{|}_{\zeta=0}^{\zeta=1}\
.\end{split}$ (135)

We now obtain an explicit expression for $S_{(3)}$ in terms of the free
solutions by using the first-order solutions, and simplifying using the
boundary limits of the bulk-to-bulk propagator (Eq. (112)) to get

$\begin{split}S_{(3)}&=ir^{d}\int_{k_{1}}\int_{k}(2\pi)^{d}\delta^{(d)}(k_{1}+k)\overline{\Psi}_{(0)}(\zeta,k_{1})\mathbb{P}_{-}\oint_{\zeta_{0}}\mathbb{S}(\zeta|\zeta_{0},k)\mathfrak{J}_{(1)}(\zeta_{0},k)\Big{|}_{\zeta=0}^{\zeta=1}\\\
&=-i\int_{k_{1,2,3}}\oint_{\zeta_{0}}\bigg{[}\overline{\psi}_{{}_{\text{R}}}(k_{1})\bar{s}_{{}_{\text{R}}}(\zeta_{0},k_{1})-\overline{\psi}_{{}_{\text{L}}}(k_{1})\bar{s}_{{}_{\text{L}}}(\zeta_{0},k_{1})\bigg{]}\phi_{(0)}(\zeta_{0},k_{2})\Psi_{(0)}(\zeta_{0},k_{3})\\\
&=-i\int_{k_{1,2,3}}\oint_{\zeta}\overline{\Psi}_{(0)}(\zeta,k_{1})\phi_{(0)}(\zeta,k_{2})\Psi_{(0)}(\zeta,k_{3})\
,\end{split}$ (136)

where we have employed the notation for momentum integrals introduced in Eq.
(50).

The correction that is quadratic in $\lambda$ or quartic in the boundary
sources takes the form

$S_{(4)}=\frac{i}{2}\oint\mathop{}\\!\mathrm{d}\zeta\mathop{}\\!\mathrm{d}^{d}\mathbf{x}\sqrt{-g}\
\phi_{(1)}\overline{\Psi}_{(0)}\Psi_{(0)}+ir^{d}\int\frac{\mathop{}\\!\mathrm{d}^{d}k}{(2\pi)^{d}}\overline{\Psi}_{(0)}(-k)\mathbb{P}_{{}_{-}}\Psi_{(2)}(k)\Big{|}_{\zeta=0}^{\zeta=1}\
.$ (137)

Using the bulk sources in Eq. (46) and Eq. (47), we obtain

$\begin{split}S_{(4)}=\
&\frac{1}{2}\int_{k_{1,2,3,4}}\oint_{\zeta_{1}}\oint_{\zeta_{2}}\overline{\Psi}_{(0)}(\zeta_{2},k_{1})\Psi_{(0)}(\zeta_{2},k_{2})\mathbb{G}(\zeta_{2}|\zeta_{1},k_{3}+k_{4})\overline{\Psi}_{(0)}(\zeta_{1},k_{3})\Psi_{(0)}(\zeta_{1},k_{4})\\\
&-i\int_{k_{1,2,3,4}}\oint_{\zeta_{1}}\oint_{\zeta_{2}}\phi_{(0)}(\zeta_{2},k_{1})\overline{\Psi}_{(0)}(\zeta_{2},k_{2})\mathbb{S}(\zeta_{2}|\zeta_{1},k_{3}+k_{4})\phi_{(0)}(\zeta_{1},k_{3})\Psi_{(0)}(\zeta_{1},k_{4})\
.\end{split}$ (138)

##  Appendix E Four-point influence phase in the past-future basis

In this appendix, we will give the explicit results for all the terms present
in the four-point Yukawa influence phase.

### Terms with bulk scalar exchange

In what follows, we will use the notation introduced in Eq. (78) to avoid
clutter. The terms in the column vector representation are:

$\begin{split}&S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}},\mathbb{G}}=-\int_{k_{1,2,3,4}}\Big{\\{}\langle\overline{S}^{\text{in}}S^{\text{in}}\mathbb{G}_{\text{adv}}\overline{S}^{\text{in}}S^{\text{out}}\rangle\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{3})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{4})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{4})\right]\\\ &\hskip
56.9055pt+\langle\overline{S}^{\text{in}}S^{\text{in}}\mathbb{G}_{\text{adv}}\overline{S}^{\text{out}}S^{\text{in}}\rangle\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{3})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{3})\right]\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{4})\Big{\\}}\ ,\end{split}$ (139)

$\begin{split}&S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}},\mathbb{G}}=-\int_{k_{1,2,3,4}}\\\
&\times\Big{\\{}\langle\overline{S}^{\text{in}}S^{\text{out}}\mathbb{G}_{\text{adv}}\overline{S}^{\text{out}}S^{\text{out}}\rangle\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{2})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{3})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{4})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{3})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{4})\right]\\\
&+\langle\overline{S}^{\text{out}}S^{\text{in}}\mathbb{G}_{\text{adv}}\overline{S}^{\text{out}}S^{\text{out}}\rangle\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{3})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{4})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{3})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{4})\right]\Big{\\}}\ ,\end{split}$ (140)

$\begin{split}&S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}},\mathbb{G}}=\int_{k_{1,2,3,4}}\langle\overline{S}^{\text{in}}S^{\text{in}}\mathbb{G}_{\text{adv}}\overline{S}^{\text{out}}S^{\text{out}}\rangle\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{3})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{4})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{3})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{4})\right]\ ,\end{split}$ (141)

and

$\begin{split}&S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}},\mathbb{G}}=\int_{k_{1,2,3,4}}\Big{\\{}\bar{\rho}^{(1)}_{\mathbb{G}}[21][43]\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{2})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{3})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{4})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{4})\right]\\\ &\hskip
85.35826pt-\bar{\rho}^{(1)}_{\mathbb{G}}[43][21]\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{3})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{4})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{4})\right]\\\ &\hskip
85.35826pt+\bar{\rho}^{(2)}_{\mathbb{G}}[12][43]\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{3})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{4})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{4})\right]\\\ &\hskip
85.35826pt-\bar{\rho}^{(2)}_{\mathbb{G}}[43][12]\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{3})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{4})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{4})\right]\\\ &\hskip
85.35826pt+\bar{\rho}^{(3)}_{\mathbb{G}}[21][34]\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{2})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{3})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{3})\right]\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{{}_{(1)}}(\bar{k}_{4})\\\
&\hskip 85.35826pt-\bar{\rho}^{(3)}_{\mathbb{G}}[34][21]\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{3})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{3})\right]\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{{}_{(1)}}(\bar{k}_{4})\\\
&\hskip 85.35826pt+\bar{\rho}^{(4)}_{\mathbb{G}}[12][34]\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{3})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{3})\right]\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{{}_{(1)}}(\bar{k}_{4})\\\
&\hskip 85.35826pt-\bar{\rho}^{(4)}_{\mathbb{G}}[34][12]\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{3})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{3})\right]\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{4})\Big{\\}}\ ,\end{split}$ (142)

where we have defined

$\begin{split}\bar{\rho}^{(1)}_{\mathbb{G}}[43][21]&\equiv\
n^{{}_{\text{BE}}}_{k^{0}_{3}+k^{0}_{4}}\langle\overline{S}^{\text{in}}S^{\text{out}}\mathbb{G}_{\text{adv}}\overline{S}^{\text{in}}S^{\text{out}}\rangle\
,\quad\bar{\rho}^{(1)}_{\mathbb{G}}[21][43]\equiv e^{-\beta k^{0}_{2}-\beta
k^{0}_{1}}\bar{\rho}^{(1)}_{\mathbb{G}}[43][21]\ ,\\\
\bar{\rho}^{(2)}_{\mathbb{G}}[43][12]&\equiv\
n^{{}_{\text{BE}}}_{k^{0}_{3}+k^{0}_{4}}\langle\overline{S}^{\text{out}}S^{\text{in}}\mathbb{G}_{\text{adv}}\overline{S}^{\text{in}}S^{\text{out}}\rangle\
,\quad\bar{\rho}^{(2)}_{\mathbb{G}}[12][43]=e^{-\beta k^{0}_{2}-\beta
k^{0}_{1}}\bar{\rho}^{(2)}_{\mathbb{G}}[43][12]\ ,\\\
\bar{\rho}^{(3)}_{\mathbb{G}}[34][21]&\equiv\
n^{{}_{\text{BE}}}_{k^{0}_{3}+k^{0}_{4}}\langle\overline{S}^{\text{in}}S^{\text{out}}\mathbb{G}_{\text{adv}}\overline{S}^{\text{out}}S^{\text{in}}\rangle\
,\quad\bar{\rho}^{(3)}_{\mathbb{G}}[21][34]=e^{-\beta k^{0}_{2}-\beta
k^{0}_{1}}\bar{\rho}^{(3)}_{\mathbb{G}}[34][21]\ ,\\\
\bar{\rho}^{(4)}_{\mathbb{G}}[34][12]&\equiv\
n^{{}_{\text{BE}}}_{k^{0}_{3}+k^{0}_{4}}\langle\overline{S}^{\text{out}}S^{\text{in}}\mathbb{G}_{\text{adv}}\overline{S}^{\text{out}}S^{\text{in}}\rangle\
,\quad\bar{\rho}^{(4)}_{\mathbb{G}}[12][34]=e^{-\beta k^{0}_{2}-\beta
k^{0}_{1}}\bar{\rho}^{(4)}_{\mathbb{G}}[34][12]\ .\end{split}$ (143)

### Terms with bulk spinor exchange

Again for notational convenience, as we did in the case of bulk scalar
exchange, we define

$\begin{split}\langle
G^{x_{1}}\overline{S}^{x_{2}}\mathbb{S}_{x}G^{x_{3}}S^{x_{4}}\rangle&\equiv\int_{\text{ext}_{1}}\int_{\text{ext}_{2}}G^{x_{1}}(\zeta_{2},k_{1})\left[\overline{S}^{x_{2}}(\zeta_{2},k_{2})\right]^{\alpha}_{\
\scalebox{1.2}{$\scriptscriptstyle\bbalpha$}}\left[\mathbb{S}_{x}(\zeta_{2}|\zeta_{1},k_{3}+k_{4})\right]^{\scalebox{1.2}{$\scriptscriptstyle\bbalpha$}}_{\
\scalebox{1.2}{$\scriptscriptstyle\bbbeta$}}\\\ &\hskip 56.9055pt\times
G^{x_{3}}(\zeta_{1},k_{3})\left[S^{x_{4}}(\zeta_{1},k_{4})\right]^{\scalebox{1.2}{$\scriptscriptstyle\bbbeta$}}_{\
\delta}\boldsymbol{J}(k_{1})\otimes\left[\boldsymbol{\overline{\psi}}(k_{2})\right]_{\alpha}\otimes\boldsymbol{J}(k_{3})\otimes\left[\boldsymbol{\psi}(k_{4})\right]^{\beta}\
,\end{split}$ (144)

where $x_{1},x_{2},x_{3},x_{4}\in\\{\text{in},\text{out}\\}$ and
$x\in\\{\text{ret},\text{adv}\\}$. Then, all the terms in the column vector
representation are:

$\begin{split}S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle P$}},\mathbb{S}}&=\
i\int_{k_{1,2,3,4}}\Big{\\{}\langle
G^{\text{in}}\overline{S}^{\text{in}}\mathbb{S}_{\text{adv}}G^{\text{in}}S^{\text{out}}\rangle\cdot
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{3})\otimes
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{3})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{4})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{4})\right]\\\ &\hskip 51.21504pt+\langle
G^{\text{in}}\overline{S}^{\text{in}}\mathbb{S}_{\text{adv}}G^{\text{out}}S^{\text{in}}\rangle\cdot
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes\left[e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{3})-e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{3})\right]\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{{}_{(1)}}(\bar{k}_{4})\\\
&\hskip 51.21504pt+\langle
G^{\text{in}}\overline{S}^{\text{out}}\mathbb{S}_{\text{ret}}G^{\text{in}}S^{\text{in}}\rangle\cdot
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{2})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{2})\right]\otimes
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{3})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{{}_{(1)}}(\bar{k}_{4})\\\
&\hskip 51.21504pt+\langle
G^{\text{out}}\overline{S}^{\text{in}}\mathbb{S}_{\text{ret}}G^{\text{in}}S^{\text{in}}\rangle\cdot\left[e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{1})-e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(1)}}(\bar{k}_{1})\right]\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{2})\otimes
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{3})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{4})\Big{\\}}\ ,\end{split}$ (145)

$\begin{split}&S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle P$}},\mathbb{S}}=\
i\int_{k_{1,2,3,4}}\\\ &\times\Big{\\{}\langle
G^{\text{in}}\overline{S}^{\text{out}}\mathbb{S}_{\text{adv}}G^{\text{out}}S^{\text{out}}\rangle\cdot
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{(1)}(\bar{k}_{1})\otimes\left(f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(2)}(\bar{k}_{2})\otimes e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(2)}(\bar{k}_{3})\otimes f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(2)}(\bar{k}_{4})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(1)}(\bar{k}_{2})\otimes e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(1)}(\bar{k}_{3})\otimes f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(1)}(\bar{k}_{4})\right)\\\ &+\langle
G^{\text{out}}\overline{S}^{\text{in}}\mathbb{S}_{\text{adv}}G^{\text{out}}S^{\text{out}}\rangle\cdot
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{(1)}(\bar{k}_{2})\otimes\left(e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(2)}(\bar{k}_{1})\otimes e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(2)}(\bar{k}_{3})\otimes f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(2)}(\bar{k}_{4})-e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(1)}(\bar{k}_{1})\otimes e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(1)}(\bar{k}_{3})\otimes f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(1)}(\bar{k}_{4})\right)\\\ &+\langle
G^{\text{out}}\overline{S}^{\text{out}}\mathbb{S}_{\text{ret}}G^{\text{in}}S^{\text{out}}\rangle\cdot
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{(1)}(\bar{k}_{3})\otimes\left(e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(2)}(\bar{k}_{1})\otimes f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(2)}(\bar{k}_{2})\otimes f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(2)}(\bar{k}_{4})-e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(1)}(\bar{k}_{1})\otimes f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(1)}(\bar{k}_{2})\otimes f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(1)}(\bar{k}_{4})\right)\\\ &+\langle
G^{\text{out}}\overline{S}^{\text{out}}\mathbb{S}_{\text{ret}}G^{\text{out}}S^{\text{in}}\rangle\cdot\left(e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(2)}(\bar{k}_{1})\otimes f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(2)}(\bar{k}_{2})\otimes e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(2)}(\bar{k}_{3})-e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(1)}(\bar{k}_{1})\otimes f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(1)}(\bar{k}_{2})\otimes e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(1)}(\bar{k}_{3})\right)\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{(1)}(\bar{k}_{4})\Big{\\}}\
,\end{split}$ (146)

$\begin{split}S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle P$}},\mathbb{S}}&=\
-i\int_{k_{1,2,3,4}}\\\ &\times\Big{\\{}\langle
G^{\text{in}}\overline{S}^{\text{in}}\mathbb{S}_{\text{adv}}G^{\text{out}}S^{\text{out}}\rangle\cdot
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{(1)}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{(1)}(\bar{k}_{2})\otimes\left(e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(2)}(\bar{k}_{3})\otimes f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(2)}(\bar{k}_{4})-e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(1)}(\bar{k}_{3})\otimes f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(1)}(\bar{k}_{4})\right)\\\ &+\langle
G^{\text{out}}\overline{S}^{\text{out}}\mathbb{S}_{\text{ret}}G^{\text{in}}S^{\text{in}}\rangle\cdot\left(e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(2)}(\bar{k}_{1})\otimes f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(2)}(\bar{k}_{2})-e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(1)}(\bar{k}_{1})\otimes f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{(1)}(\bar{k}_{2})\right)\otimes
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{(1)}(\bar{k}_{3})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle F$}}}^{(1)}(\bar{k}_{4})\Big{\\}}\
,\end{split}$ (147)

and

$\begin{split}&S_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}},\mathbb{S}}=i\int_{k_{1,2,3,4}}\Big{\\{}\bar{\rho}^{(1)}_{\mathbb{S}}[21][43]\cdot
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{1})\otimes
f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{2})\otimes
e_{\bar{\scalebox{0.9}{$\scriptscriptstyle
F$}}}^{{}_{(1)}}(\bar{k}_{3})\otimes\left[f_{\bar{\scalebox{0.9}{$\scriptscriptstyle
P$}}}^{{}_{(2)}}(\bar{k}_{4})-f_{\bar{\scalebox{0.9}{$\scriptscriptstyle