††institutetext:  1Department of Physics, McGill University, 3600 Rue
University, Montréal, H3A 2T8, QC Canada
2Department of Mathematical Sciences, Durham University, Durham, DH1 3LE,
United Kingdom

# Bootstrapping Witten diagrams via differential representation in Mellin
space

Yue-Zhou Li1, Jiajie Mei2<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We explore the use of the differential representation of AdS amplitudes to
compute Witten diagrams. The differential representation expresses AdS
amplitudes in terms of conformal generators acting on contact Witten diagrams,
which allows us to construct differential equations for Witten diagrams. These
differential equations can then be transformed into difference equations in
Mellin space, which can be solved recursively. Using this method, we
efficiently re-computed scalar four-point amplitudes and obtained new results
for scalar six-point amplitudes mediated by gluons and scalars, as well as two
examples of scalar eight-point amplitudes from gluon exchange.

##  1 Introduction

The AdS/CFT correspondence Maldacena:1997re  establishes a conjectural map to
identify the gravitational physics in Anti de-Sitter space (AdS) with
conformal field theories (CFTs) on the boundary of AdS, providing a potential
Ultra-Violet (UV) completion of AdS gravity in terms of the Hilbert space of
CFTs Witten:1998qj .

The AdS/CFT correspondence has been extensively studied, particularly in the
regime where weakly coupled and local gravitational interactions can be
described equivalently by CFTs with a large-$N$ limit and a sparse spectrum
gap Heemskerk:2009pn . In this regime, AdS observables, such as scattering
amplitudes, can be evaluated using perturbative techniques in quantum field
theory. This is accomplished by generalizing Feynman diagrams to Witten
diagrams Witten:1998qj . On the CFT side, these amplitudes correspond to the
conformal correlation functions in the large-$N$ limit Freedman:1998tz .
Therefore, the scattering amplitudes in AdS, as represented by Witten
diagrams, can shed light on understanding universal structures of large-$N$
CFTs. For example, the four-point amplitude of graviton exchange can be used
to build the leading Regge trajectory of large-$N$ CFT Costa:2012cb . Witten
diagrams can also serve as the expanding blocks of conformal correlators,
known as the Polyakov-Regge block, even beyond the large-$N$ limit
Polyakov:1974gs ; Mazac:2019shk ; Sleight:2019ive . The study of Polyakov-
Regge blocks can benefit the analytic conformal bootstrap program, as seen in
works such as Gopakumar:2016wkt ; Caron-Huot:2020adz ; Dey:2016mcs ;
Dey:2017fab ; Dey:2017oim ; Gopakumar:2018xqi .

Explicit evaluation of Witten diagrams, even at the tree-level, is notoriously
challenging due to the intricate AdS integrals involved. For instance, even
the simplest $\phi^{4}$ contact diagram in position space is expressed by a
complex function known as the $D$-function DHoker:1999kzh ; DHoker:2002nbb .
Representing AdS amplitudes through Mellin space is a more natural approach
Penedones:2010ue . For example, the $D$-function simplifies to a constant in
Mellin space. Nevertheless, computing tree-level exchange diagrams in Mellin
space using standard techniques is feasible but still remains challenging,
particularly for spinning exchange and higher-point amplitudes Paulos:2011ie ;
Fitzpatrick:2011ia .

It is noteworthy that Mellin amplitudes are a natural counterpart to flat-
space amplitudes for various reasons. Mellin amplitudes are rational functions
of Mellin variables, which are analogous to the flat-space Mandelstam
variables from a kinematic perspective. The pole structures of Mellin
amplitudes encode the exchanged operators, and factorization applies to the
corresponding residues Goncalves:2014rfa ; Fitzpatrick:2011hu , similar to the
analytic structures of flat-space amplitudes. The scaling limit of Mellin
variables enables reconstructing the flat-space amplitudes Penedones:2010ue ;
Paulos:2016fap , thus realizing the flat-space limit Polchinski:1999ry ;
Giddings:1999jq ; Gary:2009ae ; Gary:2009mi ; Heemskerk:2009pn . Therefore, it
is natural to speculate that there exists a way to efficiently compute Mellin
amplitudes by imposing correct analytic structures and consistency with the
flat-space limit, i.e., bootstrapping the Mellin amplitudes. Such a strategy
turns out to work excellently for supersymmetric correlators, as supersymmetry
imposes stronger consistency conditions. See, e.g., Rastelli:2016nze ;
Rastelli:2017udc ; Rastelli:2017ymc ; Zhou:2018ofp ; Goncalves:2019znr ;
Goncalves:2023oyx . This paper explores how to realize a similar idea for non-
supersymmetric amplitudes.

One crucial step in understanding AdS amplitudes is to make the flat-space
structures explicit. This can be achieved using the recently developed
differential representation Herderschee:2022ntr ; Cheung:2022pdk . The
differential representation expresses amplitudes as differential operators
acting on a contact diagram. This approach allows for lifting the flat-space
amplitudes to AdS schematically Eberhardt:2020ewh ; Roehrig:2020kck ;
Diwakar:2021juk ; Herderschee:2022ntr ; Gomez:2021qfd ; Gomez:2021ujt ;
Cheung:2022pdk ; Li:2022tby . The differential representation in position
space yields the Casimir differential equation for Witten diagrams
DHoker:1999kzh ; Binder:2020raz ; Zhou:2018sfz ; DHoker:1999mqo ;
Bissi:2022mrs . Fourier-transforming this differential equation to momentum
space yields the bootstrap equation for cosmological correlators presented in
Arkani-Hamed:2018kmz ; Baumann:2022jpr  and sheds light on the new structure
of correlators in momentum space Armstrong:2022csc ; Armstrong:2022mfr ;
Armstrong:2022vgl .

In this paper, we present a novel approach to efficiently compute AdS
amplitudes utilizing the differential representation in Mellin space. While
our focus is on scalar amplitudes, we allow for spinning particles to be
exchanged. Our bootstrapping strategy, as previously highlighted, combines the
correct analytic structures of Mellin amplitudes with the differential
representation. Specifically, given a Witten diagram, we provide a general
ansatz for the Mellin amplitudes with poles encoding single-trace operators.
The differential representation constructs difference equations, enabling us
to determine the coefficients in our ansatz and consequently solve for AdS
amplitudes efficiently. This approach allows us to quickly reproduce four-
point scalar amplitudes with scalar, gluon, and graviton exchange, for any
spacetime and scaling dimension. Using our bootstrap method, we also compute
higher-point amplitudes, such as six-point scalar amplitudes with gluon
exchange, and two examples of eight-point amplitudes. These amplitudes were
previously challenging to compute using standard techniques Paulos:2011ie ;
Fitzpatrick:2011ia .

This paper is organized as follows. In Section 2, we begin by providing a
brief review of the embedding formalism, Witten diagrams, and the definition
of Mellin amplitudes. We then explain how the Casimir equations for Witten
diagrams remove the single-trace operator contribution. In Section 3, we
outline an algorithm for computing AdS amplitudes in Mellin space and present
several examples of four-point amplitudes. In Section 4, we extend the
algorithm to higher-point Mellin amplitudes and include several examples such
as a six-point snowflake with scalar exchanges, a six-point snowflake with
gluon exchanges, and several truncated six-point amplitudes. We also provide
two simple examples of eight-point amplitudes. In Section 5, we offer further
remarks on the relation between our work and cosmological correlators as well
as the flat-space limit. Finally, in Appendix A, we explicitly derive the
boundary Feynman rules involved in graviton exchange and summarize the Feynman
rules for future reference. In Appendix B, we review the scalar Mellin Feynman
rules.

##  2 Generalities

###  2.1 The embedding formalism

In this paper, we work with the embedding formalism of Euclidean AdS, see
Costa:2014kfa  for more details. We use $Y$ to denote the coordinates of $d+2$
dimensional Minkowski space $\mathbb{M}^{d+2}$, which is referred to as the
embedding space. and AdSd+1 is defined by a hyperboloid subjecting to

$\displaystyle Y^{2}=-R_{\rm AdS}^{2}\,.$ (1)

We take $R_{\rm AdS}=1$ without further notice. It is instructive to think of
Euclidean AdSd+1 as a coset space ${\rm SO}(d+1,1)/{\rm SO}(d+1)$, so as to
apply harmonic analysis and convert to the isometric frame for analyzing
kinematics Cheung:2022pdk . Fields of AdSd+1 can be classified by finite-
dimensional irrep of ${\rm SO}(d+1)$ in the embedding space, transverse to the
hyperboloid. We are mostly interested in the spin-$J$ traceless symmetric
tensors and it benefits to represent them as index-free polynomials in
embedding polarizations $W$

$\displaystyle F(Y,W)=W^{A_{1}}\cdots W^{A_{J}}F_{A_{1}\cdots A_{J}}\,,$ (2)

subject to

$\displaystyle Y\cdot W=W^{2}=0\,.$ (3)

It is not hard to generalize to mixed-symmetric tensors by using different $W$
for different rows in Young diagram Costa:2018mcg ; see also Binder:2020raz
for the generalization to spinor fields. By this construction, the
contractions between two tensors in the index-free notation can be written by
summing over polarizations, namely

$\displaystyle F_{A_{1}\cdots A_{J}}H^{A_{1}\cdots
A_{J}}=\sum_{W}F(W^{\ast})H(W)={\frac{1}{J!({\frac{d-1}{2}})_{J}}}F(K^{W})H(W)\,,$
(4)

where the differential operator $K_{A}$ is to restore the indices from
polynomials Costa:2014kfa

$\displaystyle K_{A}^{W}=\big{(}{\frac{\partial}{\partial
W^{A}}}+Y_{A}(Y\cdot{\frac{\partial}{\partial
W}})\big{)}\big{(}{\frac{d-3}{2}}+W\cdot{\frac{\partial}{\partial
W}}\big{)}-{\frac{1}{2}}W_{A}\big{(}{\frac{\partial^{2}}{\partial
W^{2}}}+(Y\cdot{\frac{\partial}{\partial W}})^{2}\big{)}\,.$ (5)

We also need the covariant AdS derivatives acting on symmetric traceless
tensors encoded in polynomials of $W$ Costa:2014kfa

$\displaystyle\nabla_{A}^{Y}={\frac{\partial}{\partial
Y}}+Y_{A}\big{(}Y\cdot{\frac{\partial}{\partial
Y}}\big{)}+W_{A}\big{(}Y\cdot{\frac{\partial}{\partial W}}\big{)}\,.$ (6)

The boundary of AdS can also be embedded into $\mathbb{M}^{d+1}$ as a
lightcone surface, known as the embedding formalism for CFT Costa:2011mg ,
where we denote the coordinate by $X$ and polarization as $Z$ subject to

$\displaystyle X^{2}=X\cdot Z=Z^{2}=0\,,$ (7)

with “gauge” redundancy $Z\simeq Z+\\#X$.

Using the embedding formalism, the spin-$J$ symmetric traceless bulk-to-
boundary propagator $\Pi_{\Delta,J}(Y,W;X,Z)$ with conformal dimension
$\Delta$ satisfies the equation

$\displaystyle\big{(}(\nabla^{Y})^{2}-\Delta(\Delta-d)+J\big{)}\Pi_{\Delta,J}(Y,W;X,Z)=0\,,$
(8)

which is solved by

$\displaystyle\Pi_{\Delta,J}(Y,W;X,Z)=\mathcal{C}(\Delta,J){\frac{\big{(}2(W\cdot
X)(Z\cdot Y)-2(X\cdot Y)(W\cdot Z)\big{)}^{J}}{(-2X\cdot Y)^{\Delta+J}}}\,,$
(9)

where

$\displaystyle\mathcal{C}(\Delta,J)={\frac{(J+\Delta-1)\Gamma(\Delta)}{2\pi^{{\frac{d}{2}}}(\Delta-1)\Gamma(\Delta+1-{\frac{d}{2}})}}\,.$
(10)

In this paper, we only study scattering between external scalars, where only
$\Pi_{\Delta,0}$ and its derivatives are involved.

###  2.2 Witten diagrams and the bulk Feynman rules

In this section, we review the derivation of Witten diagrams from wick
contraction in AdS/CFT for external scalars. The essential formula is the
holographic dictionary Witten:1998qj

$\displaystyle Z_{\rm bulk}=\int[D\Phi]e^{-S[\Phi]}=\langle
e^{\int\sum_{i}\phi_{i}^{(0)}\mathcal{O}_{i}}\rangle_{\rm CFT}\,,$ (11)

where $\Phi$ denotes all possible fields in the bulk, $\phi^{(0)}_{i}$ is the
boundary value of external scalar $\phi_{i}$ whereas $\mathcal{O}_{i}$ is the
dual primary operator. The CFT correlator can then be evaluated by

$\displaystyle\langle\mathcal{O}_{1}\cdots\mathcal{O}_{n}\rangle=\Big{(}\prod_{i}{\frac{\delta}{\delta\phi_{i}^{(0)}}}\Big{)}\langle
e^{\int\sum_{i}\phi_{i}^{(0)}\mathcal{O}_{i}}\rangle_{\rm
CFT}=\Big{(}\prod_{i}{\frac{\delta}{\delta\phi_{i}^{(0)}}}\Big{)}\langle
e^{-S_{\rm int}[\Phi]}\rangle_{\rm bulk}\,,$ (12)

where the bulk expectation value is performed for free theory. The functional
derivatives $\delta/\delta\phi^{(0)}$ essentially replace bulk field $\phi$ by
the corresponding bulk-to-boundary propagator because

$\displaystyle\phi_{i}(Y)=\int
d^{d+2}X\Pi_{\Delta_{i}}(Y,X)\phi^{(0)}_{i}(X)\,.$ (13)

Then we can work on the wick contraction for the rest fields. The wick
contraction pairly groups the same (or complex conjugate, depending on the
representation of fields) bulk field contents under the bulk expectation
value, giving rise to products of bulk-to-bulk propagators (probably with bulk
derivatives). This procedure generates bulk Feynman rules.

A single Witten diagram, although not crossing symmetric, still admits the
operator product expansion (OPE) because of the conformal symmetry. A salient
property of Witten diagram is that the bulk-to-bulk propagator with conformal
dimension $\Delta$ and spin $J$ signals the existence of single-trace primary
operator $\mathcal{O}_{\Delta,J}$ in the OPE. To observe this fact, we study a
general $n+1$ point vertex with one bulk-to-bulk propagator, as shown in Fig
1.

$\cdots$$\cdot$$\cdot$$1$$2$$n$$\Phi$ Figure 1: A general $n+1$-point vertex
in a Witten diagram, where there is a bulk-to-bulk propagator with any
possible spin associated with the field $\Phi$, represented by the bold line.
The shaded region on the right denotes any possible tree-level processes that
are irrelevant to the ongoing discussion.

This vertex, as a part of a Witten diagram, can be schematically written down
after integration-by-parts. One example is

$\displaystyle\int
D^{d+2}Y\big{(}\prod_{i=1}^{n}(\nabla^{Y}\cdots)_{\alpha_{ij}}(W^{\ast}\cdot\nabla^{Y})^{J}\Pi_{\Delta_{i}}(Y,X_{i})\big{)}\Big{\langle}\Phi(Y,W)\cdots\Big{\rangle}\,,$
(14)

where $(\nabla^{Y}\cdots)_{\alpha_{ij}}$ denotes that there are $\alpha_{ij}$
covariant derivatives are contracted between $i$-th and $j$-th bulk-to-
boundary propagators, and $\Phi$ is a spin-$J$ symmetric traceless tensor
field. We can also change the ordering of derivatives to enumerate all
possible vertices. Without losing generality, any such term, as derivatives on
bulk-to-boundary propagators, must be the sum of functions of $X_{i}\cdot
X_{j},X_{i}\cdot W^{\ast}$ and $X_{i}\cdot Y$ in the following format

$\displaystyle
P^{\Delta_{i}}_{a_{i},b_{i},\beta_{ij}}=\prod_{i=1}^{n}{\frac{(X_{i}\cdot
W^{\ast})^{a_{i}-b_{i}}}{(X_{i}\cdot
Y)^{\Delta_{i}+a_{i}}}}\prod_{i<j}(X_{i}\cdot X_{j})^{\beta_{ij}}\,,\quad
b_{i}=\sum_{j\neq i}\beta_{ij}\,,$ (15)

This function satisfies the following identity involving the Casimir operator
and AdS Laplacian

$\displaystyle\mathcal{C}_{1\cdots
n}P^{\Delta_{i}}_{a_{i},b_{i},\beta_{ij}}=\Big{(}(\nabla^{Y})^{2}+J(J+d-1)\Big{)}P^{\Delta_{i}}_{a_{i},b_{i},\beta_{ij}}\,,$
(16)

where the quadratic Casimir operator involving $n$-points
$\mathcal{C}_{1\cdots n}$ is constructed by conformal generators $L_{i}^{AB}$

$\displaystyle\mathcal{C}_{1\cdots
n}={\frac{1}{2}}\big{(}\sum_{i=1}^{n}L_{i}^{AB}\big{)}^{2}\,,\quad
L_{i}^{AB}=-i\Big{(}X_{i}^{A}{\frac{\partial}{\partial
X_{iB}}}-X_{i}^{B}{\frac{\partial}{\partial X_{iA}}}\Big{)}\,.$ (17)

Using the identity (16), we can see that the differential operator

$\displaystyle\mathcal{D}_{1\cdots n}^{\Delta,J}=\mathcal{C}_{1\cdots
n}-C(\Delta,J)\,,\quad C(\Delta,J)=\Delta(\Delta-d)+J(J+d-2)\,,$ (18)

as acting on the Witten diagram, eliminates the bulk-to-bulk propagator in the
corresponding vertex. Because we can move the resulting differential operator
$(\nabla^{Y})^{2}-\Delta(\Delta-d)+J$ from
$P_{a_{i},b_{i},\beta_{ij}}^{\Delta_{i}}$ to the relevant bulk-to-bulk
propagator by integration-by-parts and recall the equation of motion
Costa:2014kfa

$\displaystyle\Big{(}(\nabla^{Y})^{2}-\Delta(\Delta-d)+J\Big{)}\Pi_{bb}^{\Delta,J}(Y,Y^{\prime};W,W^{\prime})=-\delta^{(d+2)}(Y-Y^{\prime})(W_{1}\cdot
W_{2})^{J}\,.$ (19)

This is saying the transverse and traceless bulk-to-bulk propagator is removed
by Casimir equation with quantum number $(\Delta,J)$, proving it contributes
single-trace operator in the OPE. Nevertheless, this is not the end of story.
Spinning fields can also contribute the longitudinal and trace part when two
bulk points they connect are the same, namely the longitudinal and trace part
contribute nontrivial contact terms via exchanging the longitudinal and trace
modes, respectively. It is rather easy to deal with this problem for gauge
fields (i.e., conserved operators on the boundary) such as gluon and graviton.
In this case, the longitudinal modes are nonphysical and we can take the
convenient gauge choice to simplify the calculations. In this paper, we take
the Feynman gauge for gluon, and the de Donder gauge for graviton. However, it
is worth noting that some Witten diagram has to be combined with others to
ensure gauge invariance (or at least to ensure consistency with our gauge
choice). Such diagram requires cautious treatments, as we will show in section
4.4.2. On the other hand, to deal with the trace modes, we can decompose the
fields into traceless part and trace part so that we have

$\displaystyle\Big{\langle}{\rm Tr}\,\Phi(Y,W)\cdots\Big{\rangle}\,,$ (20)

in the bulk Feynman rules. Notice ${\rm Tr}\,\Phi(Y,W)$ has two less $W$s and
thus carries spin $J-2$. This trace term is also removed by
$\mathcal{D}_{1\cdots}^{\Delta,J}$, because it simply shifts (16) by
$J\rightarrow J-2$ and consequently we end up with precisely the relevant
equation Sleight:2016hyl

$\displaystyle\Big{(}(\nabla^{Y})^{2}-J(J+d-1)+2\Big{)}({\rm
Tr}\,\Pi)_{bb}^{\Delta,J}(Y,Y^{\prime};W,W^{\prime})={\frac{J(J-1)(W_{1}\cdot
W_{2})^{J-2}\delta^{(d+2)}(Y-Y^{\prime})}{(d+2J-3)(d+2J-5)}}\,.$ (21)

We implement such example for graviton in detail in appendix A.1.

For contact diagram, there are no single-trace exchanges and the OPEs only
contain multi-trace operators DHoker:1999kzh . For any vertex with one bulk-
to-bulk propagator, we can decompose it into a single-trace operator and the
sum over multi-trace operators, see Fig 2.

$\cdots$$\cdot$$\cdot$$1$$2$$n$$\Phi$$=$$\cdots$$\cdot$$\cdot$$1$$2$$n$singletrace$+\sum$$\cdots$$\cdot$$\cdot$$1$$2$$n$multitrace
Figure 2: An $n+1$ point vertex with a bulk-to-bulk propagator $\Phi$ can be
decomposed into a single-trace operator and a sum of multi-trace operators. We
use the curved line to emphasize that it is NOT a Witten diagram, but rather a
decomposition into conformal blocks represented by using the curved lines. The
dashed line refers to the exchanged single-trace operator, and the double
dashed line refers to the exchanged multi-trace operators.

###  2.3 Casimir equations for Witten diagrams

Since the Casimir operator subtracting the corresponding eigenvalue can remove
the bulk-to-bulk propagator, we then have Casimir equations for determining a
single Witten diagram. The simplest example is four-point amplitude, as shown
in Fig 3.

$(\Delta,J)$$1$$2$$3$$4$ Figure 3: The general four-point Witten diagram,
where the exchanged single-trace operator has the scaling dimension $\Delta$
and spin $J$.

$\mathcal{D}_{12}^{\Delta,J}$ effectively modifies the OPE by prefactor but
essentially keeps conformal blocks intact

$\displaystyle
C(\Delta_{1}+\Delta_{2}+2n+J^{\prime},J^{\prime})-C(\Delta,J)\,.$ (22)

As a result, we arrive at an important identity

$\displaystyle\mathcal{D}_{12}W_{\Delta,J}^{1234}=\tilde{W}^{1234}_{\rm
contacts}\,,$ (23)

where $\tilde{W}$ denotes the resulting contract contributions from acting
with $\mathcal{D}_{12}$. Eq. (23) and its generalization to higher-points (see
eq. (24) below) are essential equations for us to compute Witten diagrams. The
explicit information of vertices now enter $\tilde{W}^{1234}_{\rm contacts}$
and we expect there are corresponding Feynman rules to determine what is
$\tilde{W}^{1234}_{\rm contacts}$ for given the bulk vertices. As noticed in
Cheung:2022pdk ; Herderschee:2022ntr  it is instructive to define the inverse
operator $\mathcal{D}_{12}^{-1}$ and then any tree-level exchanged diagram can
be written as $\mathcal{D}_{12}^{-1}\times(\text{Feynman rules for
vertices})$, which resembles the flat-space structure $1/s\times(\text{Feynman
rules for vertices})$. Similar for higher-point Witten diagrams, we can insert
the Casimir equation operator for each propagator that connects $n$-point
external legs via $n+1$-point vertex to obtain a new Witten diagram with one
less propagator

$\displaystyle\mathcal{D}_{12\cdots n}^{\Delta,J}W^{1\cdots
n\cdots}_{V,P}=W^{1\cdots n\cdots}_{V-1,P-1}\,.$ (24)

where $V$ and $P$ denote the number of vertices and propagators respectively.
This insertion can still be visualized as a cut that we call the Casimir cut,
see Fig 4 for an illustration.

$\cdots$$\cdot$$\cdot$$1$$2$$n$$(\Delta,J)$$\mathcal{D}_{12\cdots
n}^{\Delta,J}$$\,:=$$\cdots$$\cdot$$\cdot$$(\Delta,J)$$1$$2$$n$$\,=$$\cdots$$\cdot$$\cdot$$1$$2$$n$
Figure 4: The action of the differential operator $\mathcal{D}_{12\cdots
n}^{\Delta,J}$ is visualized as a Casimir cut, denoted by the vertical dashed
line. The Casimir cut removes the corresponding bulk-to-bulk propagator and
results in a new Witten diagram with a dotted effective vertex.

The Casimir cut is on the right opposite of the unitarity cut (i.e., the
double discontinuity Caron-Huot:2017vep ; Simmons-Duffin:2017nub ;
Meltzer:2019nbs ): the unitarity cut keeps only the single-trace while the
Casimir cut kills the single-trace. By recursively applying the Casimir cut
(24) , we can arrive at pure contact diagrams which are determined by
underlying Feynman rules.

However, this is not the end of story. It would be helpful to reformulate the
complicated contact derived from Casimir cuts as boundary differential
operators that act on known functions, e.g., $D_{\Delta_{i}}$ function
constructed by $\phi^{n}$ contact. The simplest such differential operator is
the conformal generator. We refer to the rules of writing contacts in terms of
conformal generators acting on $D_{\Delta_{i}}$ as the boundary Feynman rules.
To achieve this, we recall that AdSd+1 is the coset space ${\rm
SO}(d+1,1)/{\rm SO}(d+1)$, and thus employ the isometric frame techniques
developed in Cheung:2022pdk  to rewrite the contact diagrams. Refer to
Cheung:2022pdk  for more details. The idea is to use the Killing vectors

$\displaystyle\mathcal{K}_{AB}=\mathcal{K}_{AB}\,^{C}\partial^{Y}_{C}\,,\quad\mathcal{K}_{AB}\,^{C}=Y_{A}\delta_{B}^{C}-Y_{B}\delta_{A}^{C}\,,$
(25)

to rewrite contractions by replacing the AdS metric

$\displaystyle
g_{AB}=\eta_{AB}+Y_{A}Y_{B}=g_{CDEF}\mathcal{K}^{CD}\,_{A}\mathcal{K}^{EF}\,_{B}\,$
(26)

where $g_{CDEF}$ is the Killing metric

$\displaystyle
g_{CDEF}={\frac{1}{4}}(\eta_{CE}\eta_{DF}-\eta_{CF}\eta_{DE})\,.$ (27)

The Killing vectors can push the covariant derivatives to be the bulk Lorentz
generator and further to be the conformal generator by using isometry, namely

$\displaystyle\mathcal{K}_{BC}\,^{A}\nabla_{A}^{Y}\Pi_{\Delta}(Y,X)=-iL_{BC}\Pi_{\Delta}(Y,X)\,.$
(28)

If we are considering two-derivative theories, where any bulk-to-boundary
propagator has at most one derivative, the boundary Feynman rules are
straightforward using (28). This is the case that we mostly encounter in this
paper.

More generally, for higher derivatives, we can still move the Killing vectors
into derivatives to transform a particular derivative to the conformal
generator, trading by additional terms involving derivatives on Killing
vectors. The derivatives of Killing vectors are also contracting with Killing
vectors, which can be expanded by multiple Killing vectors together with
structure constant
$f_{ABCD}\,^{EF}=32g_{ABG}\,^{H}g_{CDH}\,^{I}g^{EF}\,_{I}\,{}^{G}$, modulo
$Y^{2}=-1$ due to the completeness111This statement is not rigorous because we
cannot prove it. Nevertheless, supporting examples demonstrate that any
remaining bulk coordinates $Y$ and AdS metric $g_{AB}$ can be eliminated by
employing specific bulk identities. Refer to Li:2022tby  for the case
involving gluons and gravitons., giving rise to terms with only conformal
generator (but less). The simplest such example is

$\displaystyle\mathcal{K}_{AB}\,^{C}\nabla^{Y}_{C}\mathcal{K}_{EF}\,^{G}-\mathcal{K}_{EF}\,^{C}\nabla^{Y}_{C}\mathcal{K}_{AB}\,^{G}=f_{ABEF}\,^{CD}\mathcal{K}_{CD}\,^{G}\,.$
(29)

We have not found a systematic way to perform this analysis, but it is not
hard to work out simple examples. For example, we find

$\displaystyle\int
D^{d+2}Y\big{(}\nabla^{Y}_{A}\nabla^{Y}_{B}\Pi_{\Delta_{1}}(Y,X_{1})\big{)}\big{(}\nabla^{YA}\nabla^{YB}\Pi_{\Delta_{2}}(Y,X_{2})\big{)}\Pi_{\Delta_{3}}(Y,X_{3})\Pi_{\Delta_{4}}(Y,X_{4})=$
$\displaystyle\Big{(}{\frac{1}{4}}(L_{1}\cdot
L_{2})^{2}+{\frac{d-1}{2}}L_{1}\cdot
L_{2}\Big{)}D_{\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}}\,.$ (30)

More examples can be found in Herderschee:2022ntr ; Armstrong:2022csc .

After having the boundary Feynman rules, we then have Casimir equations for
any single $n$-point Witten diagram, stating that Casimir equation operators
acting on certain Witten diagrams equals the conformal generators acting on
the $D$-function.

###  2.4 Mellin amplitudes

Although we can translate any Witten diagram to solving differential
equations, the differential equation is still highly nontrivial and intricate
to solve. The Mellin representation of correlator would make life much easier
and is also natural to check the flat-space limit Penedones:2010ue

$\displaystyle\langle\mathcal{O}_{1}\cdots\mathcal{O}_{n}\rangle={\frac{\mathcal{N}}{(2\pi
i)^{{\frac{n(n-3)}{2}}}}}\int[d\delta_{ij}]\prod_{i<j}{\frac{\Gamma(\delta_{ij})}{(-2X_{i}\cdot
X_{j})^{\delta_{ij}}}}M(\delta_{ij})\,,\quad\sum_{j\neq
i}\delta_{ij}=\Delta_{i}\,.$ (31)

Our convention of normalization is

$\displaystyle\mathcal{N}={\frac{\pi^{{\frac{d}{2}}}}{2}}\Gamma\Big{(}{\frac{\Delta_{\Sigma}-d}{2}}\Big{)}\prod_{i=1}^{n}{\frac{\mathcal{C}(\Delta_{i},0)}{\Gamma(\Delta_{i})}}\,,$
(32)

where $\Delta_{\Sigma}=\sum_{i=1}^{n}\Delta_{i}$. The Mellin amplitudes
$M(\delta_{ij})$ is the object we calculate in this paper. Notice that the
integral contour is along the imaginary axis of independent $\delta_{ij}$. By
deforming the contour to real axis and picking up poles in Mellin amplitudes
and the integral kernel (i.e., Gamma functions), one can get the correlation
functions in the coordinate space back.

Without really doing the integral of Mellin variables, we can still analyze
the OPE from Mellin amplitudes by looking into the position of poles for
Mellin variables. To analyze the OPE, we can scale the relevant singular
distance by $\lambda$ and count the power of $\lambda$. The power of $\lambda$
is a linear combination of $\delta_{ij}$, thus we are able to read off the
spectra involved in the OPE from the position of poles in $\delta_{ij}$. From
this analysis, one can immediately conclude that if a Mellin amplitude is
polynomials in $\delta_{ij}$, it must represent a certain contact diagram in
AdS, because the only poles are integers $\delta_{ij}\leq 0$ and those poles
are corresponding multi-twist operators. To see this, we can analyze
$\mathcal{O}_{1}\mathcal{O}_{2}\rightarrow\mathcal{O}$, which gives rise to
$\lambda^{-\delta_{ij}}\sim\lambda^{n}$ for integer $n$, perfectly matching
with OPE singularity
$(x_{12}^{2})^{{\frac{\Delta-J-(\Delta_{1}+\Delta_{2})}{2}}}$ for double-twist
operators $\Delta=\Delta_{1}+\Delta_{2}+J+2n$. Similarly, for higher point
functions, we can also analyze
$\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\rightarrow\mathcal{O}$, finding
triple-twist operators.

On the other hand, whenever there are additional poles in $M(\delta_{ij})$, we
conclude there are corresponding single-trace exchanges. For example

$\displaystyle
M(\delta_{ij})\sim{\frac{\lambda_{12\mathcal{O}}}{\delta_{12}-{\frac{\Delta_{1}+\Delta_{2}-(\Delta-J)}{2}}}}\,.$
(33)

This pole captures the leading term of single-trace operator
$\mathcal{O}_{\Delta,J}$ in $\mathcal{O}_{1}\mathcal{O}_{2}$ OPE, where
$\lambda_{12\mathcal{O}}$ is the OPE coefficient. There are also subsequent
poles in the OPE, more precisely, one has

$\displaystyle
M(\delta_{ij})\sim\sum_{n=0,1\cdots}{\frac{\lambda_{12\mathcal{O}}f_{m}}{\delta_{12}-{\frac{\Delta_{1}+\Delta_{2}-(\Delta-J+2n)}{2}}}}\,,$
(34)

where $f_{m}$ is function of $\delta_{ij}$ which may also contain poles giving
other single-trace operators. In general, Mellin amplitudes must have such
infinite sum of pole structures to fully encode single-trace operator, but for
special cases where the position of the pole coincides with double-twist
operators, the sum truncates there, i.e.,

$\displaystyle
M(\delta_{ij})\sim\sum_{n=0}^{n_{0}-1}{\frac{\lambda_{12\mathcal{O}}f_{m}}{\delta_{12}-{\frac{\Delta_{1}+\Delta_{2}-(\Delta-J+2n)}{2}}}}\,,\quad\Delta_{1}+\Delta_{2}-(\Delta-J+2n_{0})=0\,.$
(35)

This is because the Mellin space encodes all multi-twist families by Gamma
function factors. This truncation also appears for multi-twist operators.

In this paper, we reverse the OPE analysis described above. We first analyze
the OPE in the coordinate space and write down the most general ansatz of
Mellin amplitudes compatible with the OPE analysis, followed by solving the
ansatz using the Casimir equation. We will provide more details in the next
section.

##  3 Algorithm for AdS amplitudes in Mellin space

###  3.1 The difference equation and the algorithm

To compute a Witten diagram in Mellin space, we can formulate the Casimir
equation using Mellin space. For any differential operator $\mathcal{D}$, we
can define a difference operator to describe its action in the Mellin space

$\displaystyle\mathcal{D}\langle\mathcal{O}_{1}\cdots\mathcal{O}_{n}\rangle={\frac{\mathcal{N}}{(2\pi
i)^{{\frac{n(n-3)}{2}}}}}\int[d\delta_{ij}]\prod_{i<j}{\frac{\Gamma(\delta_{ij})}{(-2X_{i}\cdot
X_{j})^{\delta_{ij}}}}\widehat{\mathcal{D}}\circ M(\delta_{ij})\,.$ (36)

In this way, we write the equation (24) as

$\displaystyle\widehat{\mathcal{D}}_{12\cdots n}^{\Delta,J}\circ
M_{V,P}^{1\cdots n\cdots}(\delta_{ij})=M_{V-1,P-1}^{1\cdots
n\cdots}(\delta_{ij})\,.$ (37)

This equation is then a difference equation because as we act the differential
operators on the Mellin representation of Witten diagrams, the action
effectively shifts the Mellin variables. Below is the example for identical
four-point case with external dimension $\Delta_{\phi}$

$\displaystyle\widehat{\mathcal{C}}_{12}\circ
M(s,t)=\frac{1}{2}M(s,t)\left(-2ds-8\Delta_{\phi}^{2}+s^{2}+4\Delta_{\phi}(s+2t)-2st-2t^{2}\right)$
$\displaystyle-2\delta_{12}^{2}M(s-2,t)-2\delta_{12}^{2}M(s-2,t+2)+2\delta_{13}^{2}M(s,t+2)+2\delta_{14}^{2}M(s,t-2)\,,$
(38)

where for four-point we use simpler variables $(s,t)$

$\displaystyle\delta_{12}=\Delta_{\phi}-{\frac{s}{2}}\,,\quad\delta_{13}=\Delta_{\phi}-{\frac{u}{2}}\,,\quad\delta_{14}=\Delta_{\phi}-{\frac{t}{2}}\,.$
(39)

Recursively applying (37) to remove all propagators as we explained
previously, we end up with a difference equation where the RHS of equation is
pure polynomials in $\delta_{ij}$. In general, it is impossible to solve such
difference equation. Fortunately, for tree-level Mellin amplitudes, we know
the generic structure so that we can make most general ansatz with correct
pole structures and power in $\delta_{ij}$ for yet-to-be-solved Mellin
amplitudes by OPE analysis. The difference action must remove all poles in the
ansatz, which is constraining enough to solve the Mellin amplitudes in terms
of recursion equations up to finite number of undetermined constants. This
solution is fixed pure kinematically, thus capturing the Mellin transform of
conformal blocks, i.e., Mack polynomials. These constants encode the dynamics
and can be determined by the action of boundary Feynman rules in the Mellin
space, i.e., the LHS of the difference equation. We summarize this procedure
by following algorithm, in a bit abstract way, however, we will provide
examples of four-point function in a moment222See Zhou:2018sfz ; Bissi:2022mrs
; Ma:2022ihn  for a similar algorithm in the position space by using the
integrated vertex identities, which, however, requires that at least one
vertex is three-point. Our algorithm below may also be useful for capturing
the AdS bound states by modifying the ansatz for those tree-level Witten
diagrams with more fancy analytic structures Ma:2022ihn . We are grateful to
Xinan Zhou for pointing this out to us..

Algorithm

  * 1.

Given a Witten diagram, write down the most general ansatz of Mellin
amplitudes by clearly separating rational part and singular part

$\displaystyle M(\delta_{ij})={\frac{\text{rational
polynomials}}{\text{poles}}}+\text{rational polynomials}\,.$ (40)

The part with poles should be irreducible so that it does not give rise to
rational terms overlapping the pure polynomial part. The maximal power of
polynomials can be determined by performing dimensional analysis for the
number of derivatives in the Witten diagram.

  * 2.

Write down the difference equation for the Witten diagram given the ansatz
made in step $1$. Organize the difference equation so that the LHS is the
action by (multiple) Casimir equation operators, while the RHS is the
resulting contact terms given by the action of boundary Feynman rules.

  * 3.

Require the LHS of difference equation poses no pole and solve the resulting
recursion equations.

  * 4.

Substitute the solution of recursion equations into the difference equation to
fully solve the Mellin amplitude of given Witten diagram.

###  3.2 Examples of four-point

In this section, we apply our algorithm to reproduce well-known four-point
amplitudes with scalar, Yang-Mills and graviton exchanges Penedones:2010ue ;
Paulos:2011ie ; Costa:2014kfa , as a warm-up exercise and supplementary demo
for the algorithm.

We start with general single-trace exchange in four-point functions, see Fig
3. For three-point vertices, any contracted derivatives can be removed by
integration by parts and equations of motion, thus for spin-$J$ exchange, the
vertex carries $J$ derivatives. Thus we write down the most general ansatz, as
well-known in the literature

$\displaystyle
M_{4}^{\Delta,J}=\sum_{m=0,1\cdots}{\frac{Q_{Jm}(t)}{s-(\Delta-J)-2m}}+R_{J-1}(s,t)\,,$
(41)

where the maximal power of $Q_{Jm}$ and $R_{J-1}(s,t)$ is $J$ and $J-1$
respectively, namely

$\displaystyle Q_{Jm}(t)=\sum_{k=0}^{J}f_{Jkm}t^{k}\,,\quad
R_{J-1}(s,t)=\sum_{a+b\leq J-1}R_{ab}s^{a}t^{b}\,.$ (42)

Applying (38), we find the LHS of the difference equation is

$\displaystyle\text{LHS:}\,\quad\widehat{\mathcal{D}}_{12}^{\Delta,J}\circ
M_{4}^{\Delta,J}=\sum_{m=0,1\cdots}{\frac{\tilde{Q}_{J,m}(t)}{s-(\Delta-J)-2m}}+\widehat{\mathcal{D}}_{12}^{\Delta,J}\circ
R_{J-1}(s,t)\,,$ (43)

where

$\displaystyle\tilde{Q}_{J,m}(t)=\left(s-\Delta_{1}-\Delta_{2}\right)\left(s-\Delta_{3}-\Delta_{4}\right)\left(Q_{J,m-1}(t)+Q_{J,m-1}(t+2)\right)$
$\displaystyle-\left(s+t-\Delta_{1}-\Delta_{3}\right)\left(s+t-\Delta_{2}-\Delta_{4}\right)Q_{J,m}(t+2)-\left(t-\Delta_{2}-\Delta_{3}\right)\left(t-\Delta_{1}-\Delta_{4}\right)Q_{J,m}(t-2)$
$\displaystyle+\Big{(}2\left(\Delta(\Delta-d)+J(d+J-2)+\Delta_{3}\Delta_{4}\right)+s\left(2d-\Delta_{\Sigma}+2t\right)+\Delta_{2}\left(\Delta_{3}+\Delta_{4}\right)$
$\displaystyle+\Delta_{1}\left(2\Delta_{2}+\Delta_{3}+\Delta_{4}\right)-s^{2}+2t^{2}-2t\Delta_{\Sigma}\Big{)}Q_{J,m}(t)\,.$
(44)

This is equal to an effective contact diagram in the RHS, where the details
are encoded by the specific bulk theories we consider. For RHS, we will
provide the boundary Feynman rules for scalar, gluon and graviton exchanges
case by case shortly.

Requiring the LHS is free of poles yields

$\displaystyle\tilde{Q}_{Jm}(t)=\big{(}y_{0}(t)+y_{1}(t)s\big{)}\big{(}s-(\Delta-J)-2m\big{)}\,.$
(45)

Eliminating $y_{0},y_{1}$ we then find the following recursion equation

$\displaystyle-\left(\left(-\Delta_{1}-\Delta_{2}+2m+\tau\right)\left(-\Delta_{3}-\Delta_{4}+2m+\tau\right)\left(Q_{J,m-1}(t)+Q_{J,m-1}(t+2)\right)\right)$
$\displaystyle+\left(-\Delta_{1}-\Delta_{3}+2m+t+\tau\right)\left(-\Delta_{2}-\Delta_{4}+2m+t+\tau\right)Q_{J,m}(t+2)$
$\displaystyle+\Big{(}4m(m-d)-2\Delta_{3}\Delta_{4}-\Delta_{2}\left(\Delta_{3}+\Delta_{4}\right)-\Delta_{1}\left(2\Delta_{2}+\Delta_{3}+\Delta_{4}\right)+4J(1-J)-4J\tau$
$\displaystyle+\Delta_{\Sigma}(2m+\tau)+2t\left(\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}-2m-\tau\right)+4m\tau-2t^{2}-\tau^{2}\Big{)}Q_{J,m}(t)\equiv
0\,.$ (46)

This equation looks heavy, but it is precisely the equation for Mack
polynomials provided by Costa:2012cb 333To identify with eq. (131) in
Costa:2012cb , we simply swap $3\leftrightarrow 4$ and factorize out a
normalization factor $Q_{J,m}^{\rm
here}=\mathcal{P}_{Jm}Q_{J,m}^{\text{there}}$ where $\tau=\Delta-J$. We can
easily solve Eq. (46) spin by spin, recalling (42). Notably, for a general
$\Delta$, the solution reproduces the Mack polynomials Mack:2009mi ;
Costa:2012cb  up to normalization. This can be easily understood, as (41) with
the Mack polynomials inserted is proportional to the Mellin transform of the
conformal block $G^{\Delta,J}$, which obeys the Casimir equation

$\displaystyle\mathcal{D}^{\Delta,J}_{12}G^{\Delta,J}\equiv 0\,.$ (47)

This equation can only be satisfied if the LHS is also free of any poles. The
matching in this case can be easily achieved by requiring that the
normalization has zeros on double-twist operators to cancel the relevant poles
in the Gamma functions inside the Mellin integral. This is because the complex
integral of completely smooth and regular functions is vanishing. It is worth
noting that for those spinning operators saturating the unitary bounds, such
as conserved current and stress-tensor, the solution to the recursion equation
contains more undetermined constants. These constants can be fixed by the
matching procedure, and unsurprisingly, the results also agree with the Mack
polynomials. We will consider explicit examples for scalar, gluon, and
graviton exchanges, and we will observe this fact explicitly by studying gluon
and graviton exchanges

  * •

Scalar exchange

For scalar exchange, we consider the following Lagrangian

$\displaystyle\mathcal{L}=-{\frac{1}{2}}\sum_{i=1}^{4}\big{(}(\nabla\phi_{i})^{2}-m_{i}^{2}\phi_{i}^{2}\big{)}-{\frac{1}{2}}\big{(}(\nabla\phi)^{2}-m^{2}\phi^{2}\big{)}-g(\phi_{1}\phi_{2}\phi+\phi_{3}\phi_{4}\phi)\,.$
(48)

The boundary Feynman rules then give rise to the trivial RHS and the
difference equation is

$\displaystyle\widehat{\mathcal{D}}_{12}^{\Delta,0}\circ
M_{4}^{\Delta,0}=-g^{2}\,.$ (49)

The Mack polynomial is solved by

$Q_{0m}(t)=c\,\mathcal{P}_{0m}\,,$ (50)

where $c$ is the normalization constant and $\mathcal{P}_{0m}$ is

$\displaystyle\mathcal{P}_{Jm}=\frac{\left(\frac{1}{2}\left(\tau-\Delta_{1}-\Delta_{2}+4\right)\right)_{m-1}\left(\frac{1}{2}\left(\tau-\Delta_{3}-\Delta_{4}+4\right)\right)_{m-1}}{(2)_{m-1}\left(-\frac{d}{2}+\Delta+2\right)_{m-1}}\,.$
(51)

We substitute this solution back to (41) for $J=0$ and $R_{-1}\equiv 0$, after
performing the infinite sum over $m$ it shall match with the RHS. This
procedure gives

$\begin{split}c_{0}=g^{2}\frac{(2+\Delta-\Delta_{1}-\Delta_{2})(2+\Delta-\Delta_{3}-\Delta_{4})\Gamma(\frac{\Delta-d+\Delta_{1}+\Delta_{2}}{2})\Gamma(\frac{\Delta-d+\Delta_{3}+\Delta_{4}}{2})}{16\Gamma(2+\Delta-\frac{d}{2})\Gamma(\frac{\Delta_{\Sigma}-d}{2})}\,,\end{split}$
(52)

which perfectly matches with the result from directly doing Feynman integral
in the bulk Penedones:2010ue .

  * •

Gluon exchange

Let’s then take gluon exchange between four scalars as example. We consider
the following Lagrangian

$\displaystyle\mathcal{L}=-{\frac{1}{2}}\sum_{i=1,3}(|D\phi_{i}|^{2}-m_{i}^{2}|\phi_{i}|^{2})-{\frac{1}{4g_{\rm
YM}^{2}}}F^{a}_{\mu\nu}F^{\mu\nu a}\,,$ (53)

where $\Delta_{2}=\Delta_{1},\Delta_{4}=\Delta_{3}$. The boundary Feynman
rules for this interaction can be easily worked out Herderschee:2022ntr

$\begin{split}\widehat{\mathcal{D}}_{12}^{d-1,1}\circ
M_{4}^{d-1,1}&=f_{s}\big{(}\widehat{L}_{2}\cdot\widehat{L}_{3}-\widehat{L}_{2}\cdot\widehat{L}_{4}\big{)}\circ
1\\\
&=2f_{s}\left(d-2\left(\Delta_{1}+\Delta_{3}\right)\right)\left(\Delta_{1}+\Delta_{3}-\frac{s}{2}-t\right)\,,\end{split}$
(54)

where $f_{s}=f^{12a}f^{a34}g_{\rm YM}^{2}$ is the colour factor multiplied
with the Yang-Mills coupling. In fact, due to the antisymmetry nature of YM
interaction, we can simply replace $\widehat{L}_{i}\cdot\widehat{L}_{j}$ by
$\delta_{ij}$ up to appropriate overall factor. This property is actually
hidden by using $(s,t)$.

The single-trace operator in this case is conserved current on CFT side with
$\Delta=d-1,J=1$, and we find solution of (46) with more unfixed coefficients

$\displaystyle f_{11m}=c_{1}\mathcal{P}_{1m}\,,\quad
f_{10m}=\big{(}c_{2}\,m-\frac{1}{2}c_{1}(m-1)\left(d-2\left(\Delta_{1}+\Delta_{3}+1\right)\right)\big{)}\mathcal{P}_{1m}\,.$
(55)

To determine the coefficients $c_{i},R_{0}$, we substitute the solution back
to (54) and resum the series. In the end we find

$\begin{split}R_{0}=-f_{s}\,,\quad
c_{2}={\frac{1}{2}}c_{1}(d-2(\Delta_{1}+\Delta_{3}))\,,\quad
c_{1}=f_{s}\frac{\left(d-2\Delta_{1}\right)\left(d-2\Delta_{3}\right)\Gamma\left(\Delta_{1}\right)\Gamma\left(\Delta_{3}\right)}{4\Gamma\left(\frac{d}{2}+1\right)\Gamma\left(-\frac{d}{2}+\Delta_{1}+\Delta_{3}\right)}\,.\end{split}$
(56)

It is important to note that by this matching we solved $c_{2}$ in terms of
$c_{1}$, which precisely reproduces the Mack polynomials with
$\Delta=d-1,J=1$. After summing over all poles, it perfectly matches with
Paulos:2011ie .

  * •

Graviton exchange

We consider scalar theories minimally coupled to AdS graviton

$\displaystyle\mathcal{L}=-{\frac{1}{2}}\sum_{i=1,3}\Big{(}\nabla_{\mu}\phi_{i}\nabla_{\nu}\phi_{i}h^{\mu\nu}-{\frac{1}{2}}g_{\mu\nu}h^{\mu\nu}\big{(}(\nabla\phi)^{2}-m_{i}^{2}\phi_{i}^{2}\big{)}\Big{)}\,.$
(57)

The corresponding boundary Feynman rules gives Herderschee:2022ntr

$\displaystyle\widehat{\mathcal{D}}_{12}^{d,2}\circ M_{4}^{d,2}=-8\pi
G\Big{(}{\frac{1}{2}}\big{(}\widehat{L}_{1}\cdot\widehat{L}_{4}\,\widehat{L}_{2}\cdot\widehat{L}_{3}+\widehat{L}_{1}\cdot\widehat{L}_{3}\,\widehat{L}_{2}\cdot\widehat{L}_{4}-\widehat{L}_{1}\cdot\widehat{L}_{2}\,\widehat{L}_{3}\cdot\widehat{L}_{4}\big{)}$
$\displaystyle+\Delta_{1}\tilde{\Delta}_{1}\widehat{L}_{3}\cdot\widehat{L}_{4}+\Delta_{3}\tilde{\Delta}_{3}\widehat{L}_{1}\cdot\widehat{L}_{4}-{\frac{2\Delta_{1}\tilde{\Delta}_{1}\Delta_{3}\tilde{\Delta}_{3}(d+1)}{d-1}}\Big{)}\circ
1\,,$ (58)

where $\tilde{\Delta}=d-\Delta$ is the shadow dimension.

Since the exchanged operator is stress-tensor, similar to conserved current,
solution of (46) contains one more undetermined coefficients

$\displaystyle f_{22m}=c_{1}\mathcal{P}_{2m}\,,\quad
f_{21m}=\big{(}c_{2}\,m+c_{1}(1-m)\left(d-2\left(\Delta_{1}+\Delta_{3}+1\right)\right)\big{)}\mathcal{P}_{2m}\,,$
$\displaystyle
f_{20m}=\frac{1}{4}\Big{(}2c_{2}m\left(d-2\left(\Delta_{1}+\Delta_{3}+1\right)+2m\right)+c_{1}\Big{(}4\Delta_{1}\left(d(2m-1)+\Delta_{3}(2-4m)+2(m-1)^{2}\right)$
$\displaystyle+4\Delta_{3}\Big{(}d(2m-1)+\Delta_{3}(1-2m)+2(m-1)^{2}\big{)}-\frac{(d+2m-2)((d-2)d(2m-1)+4m)}{d-1}$
$\displaystyle+\Delta_{1}^{2}(4-8m)\Big{)}\Big{)}\mathcal{P}_{2m}\,.$ (59)

Resuming the series and solving (58) yields

$\displaystyle R_{0,0}=8\pi
G(2(\Delta_{1}^{2}+\Delta_{3}^{2}+\Delta_{1}\Delta_{3})-d(\Delta_{1}+\Delta_{3}))\,,\quad
R_{0,1}=8\pi G(d-2(\Delta_{1}+\Delta_{3}))\,,\quad R_{1,0}=0\,,$
$\displaystyle c_{1}=\frac{2\pi
G\left(d-2\Delta_{1}\right)\left(d-2\Delta_{3}\right)\Gamma\left(\Delta_{1}+2\right)\Gamma\left(\Delta_{3}+2\right)}{\left(\Delta_{1}+1\right)\left(\Delta_{3}+1\right)\Gamma\left(\frac{d}{2}+2\right)\Gamma\left(-\frac{d}{2}+\Delta_{1}+\Delta_{3}\right)}\,,\quad
c_{2}=(d-2(\Delta_{1}+\Delta_{3}))c_{1}\,.$ (60)

$c_{1}$ and $c_{2}$ solved above is also consistent with Mack polynomials with
$\Delta=d,J=2$. Our result agrees with bulk Feynman rules Costa:2014kfa  by
changing conventions $s\rightarrow t,t\rightarrow s+\Delta_{1}+\Delta_{3}$.

##  4 Higher-point Mellin amplitudes

In this section, we apply our algorithm to study higher point Mellin
amplitudes. We focus on six-point snowflake topology, which is expected to
encode building blocks for all other topologies. Since the implementation of
the algorithm for higher points is a bit more intricate than four points, we
start with the scalar exchange as a warm-up, which shall match with scalar
Mellin Feynman rules constructed in Nandan:2011wc . Following the route of the
scalar case, we then solve the most general snowflake six-point amplitude with
three gluon exchanges. Besides, we also report simplified examples (i.e., with
special conformal and spacetime dimensions) for other six-point amplitudes and
two examples for eight-point amplitudes.

###  4.1 Warm-up: six-point snowflake with scalar exchanges

Consider snowflake scalar exchange Fig 5, where we include the most general
three-point vertex $\sum g_{ijk}\phi_{i}\phi_{j}\phi_{k}$.

$\Delta_{p_{3}}$$\Delta_{p_{2}}$$\Delta_{p_{1}}$$1$$2$$3$$4$$5$$6$ Figure 5:
Six-point snowflake Witten diagram with scalar exchanges $\Delta_{p_{1}}$,
$\Delta_{p_{2}}$ and $\Delta_{p_{3}}$.

The corresponding equation is

$\displaystyle\widehat{\mathcal{D}}_{56}^{\Delta_{p_{1}},0}\widehat{\mathcal{D}}_{34}^{\Delta_{p_{2}},0}\widehat{\mathcal{D}}_{12}^{\Delta_{p_{3}},0}\circ
M_{12s,sss,s56,s34}=-g_{12,p_{3}}g_{34,p_{2}}g_{56,p_{1}}g_{p_{1},p_{2},p_{3}}\,.$
(61)

Using the algorithm, we recursively solve this equation.

First, on the top level, we aim to solve the effective amplitude induced by

$\displaystyle M^{\rm
eff}_{1234s,s56}:=\widehat{\mathcal{D}}_{34}^{\Delta_{p_{2}},0}\widehat{\mathcal{D}}_{12}^{\Delta_{p_{3}},0}\circ
M_{12s,sss,s56,s34}\,,\quad\widehat{\mathcal{D}}_{56}^{\Delta_{p_{1}},0}\circ
M^{\rm eff}_{1234s,s56}=-\widehat{g}\,$ (62)

where $\widehat{g}=g_{12,p_{3}}g_{34,p_{2}}g_{56,p_{1}}g_{p_{1},p_{2},p_{3}}$.
See Fig 6 below.

$\Delta_{p_{1}}$$1$$2$$3$$4$$5$$6$$\,=$$1$$2$$3$$4$$5$$6$ Figure 6: The
Casimir cut equation for the effective amplitudes $M^{\rm eff}_{1234s,s56}$,
where the black dot refers to the effective vertices.

This step makes no much difference from four-point case. The ansatz at this
level is

$\displaystyle M^{\rm eff}_{1234s,s56}=\sum_{m}{\frac{f^{\rm
eff}_{56,m}}{\delta_{56}+m+{\frac{\Delta_{p_{1}}-\Delta_{5}-\Delta_{6}}{2}}}}\,.$
(63)

Similar to four-point example, we solve $f^{\rm eff}_{56,m}$ by simply
requiring the cancelation of pole in $\delta_{56}$

$\displaystyle f^{\rm eff}_{56,m}=c^{\rm
eff}_{56}\mathcal{P}_{0\Delta_{p_{1}}m}^{\sum_{i=1}^{4}\Delta_{i},\Delta_{5}+\Delta_{6},\Delta_{p_{1}}}\,,$
(64)

where $\mathcal{P}_{J\tau m}^{a,b,c}$ is a slight modification of (51)

$\displaystyle\mathcal{P}_{J\tau
m}^{a,b,c}=\frac{\left(\frac{1}{2}\left(\tau-a+4\right)\right)_{m-1}\left(\frac{1}{2}\left(\tau-b+4\right)\right)_{m-1}}{(2)_{m-1}\left(-\frac{d}{2}+c+2\right)_{m-1}}\,.$
(65)

This function does not depend on $J$ explicitly, the role of $J$-subscript is
nothing but reminding us what spin is exchanged. We will see this factor is
quite universal in our discussions, and we usually factorize it out to make
the difference equation simpler to solve. Then by the top level matching, we
obtain

$\displaystyle c^{\rm
eff}_{56}=\frac{\widehat{g}\left(\Delta_{p_{1}}-\sum_{i=1}^{4}\Delta_{i}+2\right)\left(\Delta_{p_{1}}-\Delta_{5}-\Delta_{6}+2\right)\Gamma\left(\frac{\sum_{i=1}^{4}\Delta_{i}+\Delta_{p_{1}}-d}{2}\right)\Gamma\left(\frac{\Delta_{5}+\Delta_{6}+\Delta_{p_{1}}-d}{2}\right)}{16\,\Gamma\left(\frac{\Delta_{\Sigma}-d}{2}\right)\Gamma\left(-\frac{d}{2}+\Delta_{p_{1}}+2\right)}\,.$
(66)

We can easily check this result matches with scalar Mellin Feynman rules. (we
review the scalar Mellin Feynman rules in appendix B).

The next step is then to solve the equation for an effective amplitude $M^{\rm
eff}_{34s,56s,12ss}$ as shown in Fig 7

$\displaystyle M^{\rm
eff}_{12ss,s34,s56}:=\widehat{\mathcal{D}}_{12}^{\Delta_{p_{3}},0}\circ
M_{12s,sss,s56,s34}\,,\quad\widehat{\mathcal{D}}_{34}^{\Delta_{p_{2}},0}\circ
M^{\rm eff}_{12ss,s34,s56}=M^{\rm eff}_{1234s,s56}\,,$ (67)

where the ansatz is

$\displaystyle M^{\rm eff}_{12ss,s34,s56}=\sum_{n,m}{\frac{f^{\rm
eff}_{34,n,m}}{\big{(}\delta_{34}+n+{\frac{\Delta_{p_{2}}-\Delta_{3}-\Delta_{4}}{2}}\big{)}\big{(}\delta_{56}+m+{\frac{\Delta_{p_{1}}-\Delta_{5}-\Delta_{6}}{2}}\big{)}}}\,.$
(68)

$\Delta_{p_{2}}$$\Delta_{p_{1}}$$1$$2$$3$$4$$5$$6$$\,=$$\Delta_{p_{1}}$$1$$2$$3$$4$$5$$6$
Figure 7: The Casimir cut equation for the effective amplitudes $M^{\rm
eff}_{12ss,s34,s56}$.

Now it is more subtle to require the disappearance of poles in $\delta_{34}$
after the action of $\mathcal{D}_{34}^{\Delta_{p_{2}},0}$. The action of
$\mathcal{D}_{34}^{\Delta_{p_{2}},0}$ leads to a reducible expression that can
be further decomposed into irreducible partial fractions with respect to
$\delta_{56}$, namely

$\displaystyle\widehat{\mathcal{D}}_{34}^{\Delta_{p_{2}},0}\circ M^{\rm
eff}_{12ss,s34,s56}=\sum_{nm}\Big{(}{\textstyle{\frac{\scriptstyle
4\left(f^{\text{eff}}_{34,n-1,m}-f^{\text{eff}}_{34,n-1,m-1}\right)\delta_{34}}{\scriptstyle\delta_{34}+n+{\frac{\Delta_{p_{2}}-\Delta_{3}-\Delta_{4}}{2}}}}}-{\textstyle{\frac{\scriptstyle\mathcal{F}_{nm}(\delta_{34})}{\scriptstyle\big{(}\delta_{34}+n+{\frac{\Delta_{p_{2}}-\Delta_{3}-\Delta_{4}}{2}}\big{)}\big{(}\delta_{56}+m+{\frac{\Delta_{p_{1}}-\Delta_{5}-\Delta_{6}}{2}}\big{)}}}}\Big{)}\,,$
(69)

where

$\displaystyle\mathcal{F}_{nm}(\delta_{34})=(2\delta_{34}-\Delta_{3}-\Delta_{4}+\Delta_{p_{2}})(-d-2\delta_{34}+\Delta_{3}+\Delta_{4}+\Delta_{p_{2}})f^{\text{eff}}_{34,n,m}$
$\displaystyle+2\delta_{34}((2\delta_{34}+\Delta_{13}+\Delta_{24}+2m+\Delta_{p_{1}})f^{\text{eff}}_{34,n-1,m}+(\Delta_{5}+\Delta_{6}-2m-\Delta_{p_{1}})f^{\text{eff}}_{34,n-1,m-1})\,.$
(70)

Notably, the first term in (69) completely vanishes, since it allows to shift
$m-1$ to $m$ without changing the result. Then kinematically
$\mathcal{F}_{nm}(\delta_{34})$ should cancel pole in $\delta_{34}$. We
factorize an appropriate factor as follows

$\displaystyle f^{\rm
eff}_{34,nm}=\mathcal{P}_{0\Delta_{p_{2}}n}^{\Delta_{p_{1}}+\Delta_{1}+\Delta_{2}+2m,\Delta_{3}+\Delta_{4},\Delta_{p_{2}}}{\frac{\big{(}{\frac{1}{2}}(4+\Delta_{p_{1}}-\Delta_{5}-\Delta_{6})\big{)}_{m-1}}{\big{(}{\frac{1}{2}}(\Delta_{p_{1}p_{2}}+\Delta_{1}+\Delta_{2})\big{)}_{m-1}}}\widehat{f}^{\rm
eff}_{34,nm}\,.$ (71)

Then we find an extremely simple recursion equation for $\widehat{f}$

$\displaystyle\widehat{f}^{\rm eff}_{34,nm}+\widehat{f}^{\rm
eff}_{34,n-1,m-1}-\widehat{f}^{\rm eff}_{34,n-1,m}\equiv 0\,,$ (72)

which can be simply solved by

$\displaystyle\widehat{f}^{\rm
eff}_{34,nm}=\sum_{k=0}^{m}c_{34,k}{\frac{(-n)_{m-k}}{(1)_{m-k}}}\,.$ (73)

We note $\widehat{f}^{\rm eff}_{34,nm}$ is a universal building block that
also appears in other diagrams such as gluon exchange. The dynamics now are
encoded in the undetermined coefficients $c_{34,k}$, requiring efforts to do
the matching procedure. Similar to the four-point case, in the matching
procedure we have to sum infinite series for $n$, as a result, we obtain a lot
of hypergeometric functions which can be, although not manifestly, easily
reduced to simple Gamma functions. We emphasize again what we aim to match is
(66), which is itself an infinite series. The simplest way we can imagine is
to do matching $m$ by $m$, thanks to the denominator $\delta_{56}+m+\cdots$.
Not surprisingly ultimately we find a unique solution of $c_{34,k}$.

$\displaystyle c_{34,k}=-c^{\rm
eff}_{56}\,(k+1)\mathcal{P}_{0,\Delta_{p_{1}}-1,k+1}^{\Delta_{p_{2}}+1+\Delta_{1}+\Delta_{2},\Delta_{p_{2}}+3-\Delta_{1}-\Delta_{2},\Delta_{p_{1}}-1}\times$
$\displaystyle{\textstyle{\frac{\scriptstyle\left(-\Delta_{3}-\Delta_{4}+\Delta_{p_{2}}\right)\left(-\Delta_{3}-\Delta_{4}+\Delta_{p_{2}}+2\right)\left(-d+2\Delta_{p_{1}}+2\right)\Gamma\left(\frac{1}{2}\left(-d+\Delta_{3}+\Delta_{4}+\Delta_{p_{2}}+2\right)\right)\Gamma\left(\frac{1}{2}\left(-d+\Delta_{1}+\Delta_{2}+\Delta_{p_{1}}+\Delta_{p_{2}}\right)\right)}{\scriptstyle
2\left(-\Delta_{1}-\Delta_{2}-\Delta_{3}-\Delta_{4}+\Delta_{p_{1}}+2\right)\Gamma\left(\frac{1}{2}\left(-d+\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}+\Delta_{p_{1}}\right)\right)\Gamma\left(-\frac{d}{2}+\Delta_{p_{2}}+2\right)}}}\,.$
(74)

It is easy to check with (66) that the result at this level is symmetric in
$(m,\Delta_{5,6})\leftrightarrow(n,\Delta_{3,4})$, and also perfectly matches
with the scalar Mellin Feynman rules.

Finally, at the bottom level, we have equation

$\displaystyle\widehat{\mathcal{D}}_{12}^{\Delta_{p_{3}},0}\circ
M_{12s,sss,s56,s34}=M^{\rm eff}_{12ss,s34,s56}\,,$ (75)

as shown in Fig 8.

$\Delta_{p_{3}}$$\Delta_{p_{2}}$$\Delta_{p_{1}}$$1$$2$$3$$4$$5$$6$$\,=$$\Delta_{p_{1}}$$\Delta_{p_{2}}$$1$$2$$3$$4$$5$$6$
Figure 8: The Casimir cut equation for $M_{12s,sss,s56,s34}$ that defines the
effective amplitude $M^{\rm eff}_{12ss,s34,s56}$.

The ansatz as usual is given by

$\displaystyle
M_{12s,sss,s56,s34}=\sum_{pnm}{\frac{f_{12,pnm}}{\big{(}\delta_{12}+p+{\frac{\Delta_{p_{3}}-\Delta_{1}-\Delta_{2}}{2}}\big{)}\big{(}\delta_{34}+n+{\frac{\Delta_{p_{2}}-\Delta_{3}-\Delta_{4}}{2}}\big{)}\big{(}\delta_{56}+m+{\frac{\Delta_{p_{1}}-\Delta_{5}-\Delta_{6}}{2}}\big{)}}}\,.$
(76)

There is no more subtlety at this level. Similarly, we find it is instructive
to factorize $f_{12,pnm}$ to

$\displaystyle
f_{12,pnm}=\mathcal{P}_{0\Delta_{p_{3}}p}^{\Delta_{p_{1}}+\Delta_{p_{2}}+2(m+n),\Delta_{1}+\Delta_{2},\Delta_{p_{3}}}{\textstyle{\frac{\scriptstyle\left(\frac{1}{2}\left(-\Delta_{5}-\Delta_{6}+\Delta_{p_{1}}+4\right)\right)_{m-1}\left(\frac{1}{2}\left(-\Delta_{3}-\Delta_{4}+\Delta_{p_{2}}+4\right)\right)_{n-1}}{\scriptstyle\left(\frac{1}{2}\left(\Delta_{p_{1}}+\Delta_{p_{2}}-\Delta_{p_{3}}\right)\right)_{m+n-1}}}}\widehat{f}_{12,pnm}\,.$
(77)

The cancelation of $\delta_{12}$ pole then provides another universal equation

$\displaystyle\widehat{f}_{12,pnm}+\widehat{f}_{12,p-1,n,m-1}+\widehat{f}_{12,p-1,n-1,m}-\widehat{f}_{12,p-1,n,m}\equiv
0\,.$ (78)

The solution to this recursion equation is

$\displaystyle\widehat{f}_{12,pnm}=\sum_{k_{1}=0}^{n}\sum_{k_{2}=0}^{m}c_{12,k_{1}k_{2}}{\frac{(-p)_{n+m-k_{1}-k_{2}}}{(1)_{n-k_{1}}(1)_{m-k_{2}}}}\,.$
(79)

By matching to $M^{\rm eff}_{12ss,s34,s56}$, we find

$\displaystyle
c_{12,k_{1},k_{2}}=c_{34,0}\sum_{q}(-1)^{k_{1}+k_{2}+1}\left(-\Delta_{1}-\Delta_{2}+\Delta_{p_{3}}+2\right)\left(-d+2\Delta_{p_{2}}+2\right)\left(-k_{1}\right)_{q}\times$
$\displaystyle{\textstyle{\frac{\scriptstyle\left(\frac{1}{2}\left(-d+\Delta_{1}+\Delta_{2}+\Delta_{p_{1}}+\Delta_{p_{2}}\right)\right)_{k_{1}}\left(\frac{1}{2}\left(-d+2k_{1}+\Delta_{1}+\Delta_{2}+\Delta_{p_{1}}+\Delta_{p_{2}}\right)\right)_{k_{2}}\left(\frac{1}{2}\left(-d+\Delta_{p_{1}}+\Delta_{p_{2}}+\Delta_{p_{3}}\right)\right)_{q}\left(\frac{1}{2}\left(-2q+\Delta_{p_{1}p_{3}}-\Delta_{p_{3}}+2\right)\right)_{k_{2}}}{\scriptstyle
16k_{1}!k_{2}!q!\left(\frac{1}{2}\left(-d+2\Delta_{p_{1}}+2\right)\right)_{k_{2}}\left(\frac{1}{2}\left(-d+2\Delta_{p_{2}}+2\right)\right)_{q}}}}\,,$
(80)

which then gives us the complete answer of the snowflake six-point Mellin
amplitudes with scalar exchange, as expected by scalar Mellin Feynman rules.

###  4.2 Six-point snowflake with gluon exchanges

Educated by the scalar snowflake exercise, we are ready to report the
snowflake gluon result as shown in Fig 9.

$1$$2$$3$$4$$5$$6$ Figure 9: Six-point snowflake Witten diagram with gluon
exchanges. The coil lines represent gluon, giving rise to single-trace
conserved current with $(\Delta=d-1,J=1)$.

This diagram can be generated by the Lagrangian in (53) appended by additional
scalar $\phi_{5}$. As constructed in Herderschee:2022ntr  (with a slight
generalization to any mass), the boundary Feynman rule for this diagram is
(under the action
$\widehat{\mathcal{D}}_{56}^{d-1,1}\widehat{\mathcal{D}}_{34}^{d-1,1}\widehat{\mathcal{D}}_{12}^{d-1,1}\circ
M_{12g,ggg,g56,g34}$

$\displaystyle-{\frac{C_{\rm
snow}}{2}}\Big{(}\big{(}\widehat{L}_{13}\widehat{L}_{26}-\widehat{L}_{13}\widehat{L}_{25}+\widehat{L}_{13}\widehat{L}_{45}-\widehat{L}_{13}\widehat{L}_{46}+\widehat{L}_{23}\widehat{L}_{46}-\widehat{L}_{23}\widehat{L}_{45}\big{)}-\big{(}3\leftrightarrow
4\big{)}$ $\displaystyle-\big{(}3\leftrightarrow 5,4\leftrightarrow
6\big{)}-\big{(}3\leftrightarrow 6,4\leftrightarrow 5\big{)}\Big{)}\circ 1\,,$
(81)

where $C_{\rm snow}$ encodes the color factor and Yang-Mills coupling

$\displaystyle C_{\rm
snow}=f^{12g_{1}}f^{34g_{2}}f^{56g_{3}}f^{g_{1}g_{2}g_{3}}g_{\rm YM}^{4}\,.$
(82)

This action, as evaluated in Mellin space, gives rise to

$\displaystyle P_{\rm snow}=2C_{\rm
snow}\tilde{\Delta}_{\Sigma}(2-\tilde{\Delta}_{\Sigma})\Big{(}\big{(}\delta_{13}\delta_{26}-\delta_{13}\delta_{25}+\delta_{13}\delta_{45}-\delta_{13}\delta_{46}+\delta_{23}\delta_{46}-\delta_{23}\delta_{45}\big{)}-\big{(}3\leftrightarrow
4\big{)}$ $\displaystyle-\big{(}3\leftrightarrow 5,4\leftrightarrow
6\big{)}-\big{(}3\leftrightarrow 6,4\leftrightarrow 5\big{)}\Big{)}\,.$ (83)

As we already observed in four-point case, this function simply replaces
$\widehat{L}_{i}\cdot\widehat{L}_{j}$ by $\delta_{ij}$ in the boundary Feynman
rule, thanks to the gauge group. Furthermore, we emphasize that this “flat
extension” property enables us to find an extremely simple ansatz, putting
gluon and scalar exchange on the same level

$\displaystyle M^{\rm eff}_{1234g,g56}=\sum_{m}{\frac{f_{56g,m}P_{\rm
snow}}{\delta_{56}+m+{\frac{d-2-2\Delta_{5}}{2}}}}\,,$ $\displaystyle M^{\rm
eff}_{12gg,g34,g56}=\sum_{nm}{\frac{f_{34g,nm}P_{\rm
snow}}{\big{(}\delta_{34}+n+{\frac{d-2-2\Delta_{3}}{2}}\big{)}\big{(}\delta_{56}+m+{\frac{d-2-2\Delta_{5}}{2}}\big{)}}}\,,$
$\displaystyle M_{12g,ggg,g56,g34}=\sum_{pnm}{\frac{f_{12g,pnm}P_{\rm
snow}}{\big{(}\delta_{12}+p+{\frac{d-2-2\Delta_{1}}{2}}\big{)}\big{(}\delta_{34}+n+{\frac{d-2-2\Delta_{3}}{2}}\big{)}\big{(}\delta_{56}+m+{\frac{d-2-2\Delta_{5}}{2}}\big{)}}}\,,$
(84)

where $M^{\rm eff}$ is defined through Casimir cuts, similar to those in the
previous section by replacing $s$ by $g$, $J=0$ by $J=1$, specifying
$\Delta_{p_{i}}=d-1$. Generally, if we are thinking about arbitrary spin-$1$
exchange, we have to make ansatz containing the most general polynomials up to
the degree of the number of involved derivatives, which include a large number
of yet-to-be-solved functions. The nice property of YM allows us to reduce to
only one function in the ansatz.

The top level makes no much difference from the scalar exchange, we arrive at

$\displaystyle f^{\rm eff}_{56g,m}=c_{56g}^{\rm
eff}\mathcal{P}_{1,d-1,m}^{2(\Delta_{1}+\Delta_{3}+1),2\Delta_{5},d-1}\,,$
(85)

for which the matching now trivially determines

$\displaystyle c_{56g}^{\rm
eff}=-\frac{\left(d-2\Delta_{1}-2\Delta_{3}-2\right)\left(d-2\Delta_{5}\right)\Gamma\left(\Delta_{1}+\Delta_{3}+1\right)\Gamma\left(\Delta_{5}\right)}{16\Gamma\left(\frac{d}{2}+1\right)\Gamma\left(-\frac{d}{2}+\Delta_{1}+\Delta_{3}+\Delta_{5}+2\right)}\,.$
(86)

At the next level, we obtain

$\displaystyle f^{\rm
eff}_{34g,nm}=\mathcal{P}_{1,d-2,n}^{d+2\Delta_{1}+2m,2\Delta_{3},d-1}{\frac{\big{(}1+{\frac{d}{2}}-\Delta_{5})\big{)}_{m-1}}{\big{(}1+\Delta_{1})\big{)}_{m-1}}}\widehat{f}^{\rm
eff}_{34,nm}\,,$ (87)

where $\widehat{f}^{\rm eff}_{34,nm}$ satisfies eq. (72) and is solved by (73)
by replacing $c^{\rm eff}_{34,k}$ by $c^{\rm eff}_{34g,k}$

$\displaystyle c^{\rm eff}_{34g,k}=c^{\rm
eff}_{56g}(k+1)\mathcal{P}_{1,d-2,k+1}^{d+2+2\Delta_{1},d-2\Delta_{1},d-2}\times\frac{\left(d-2\Delta_{3}\right)\Gamma\left(\Delta_{3}\right)\Gamma\left(\frac{d}{2}+\Delta_{1}\right)}{4\left(\Delta_{1}+\Delta_{3}\right)\left(d-2\left(\Delta_{1}+\Delta_{3}+1\right)\right)\Gamma\left(\frac{d}{2}\right)\Gamma\left(\Delta_{1}+\Delta_{3}\right)}\,.$
(88)

Finally, we move to the bottom level and obtain the solution of the target
amplitude

$\displaystyle
f_{12,pnm}=\mathcal{P}_{0,d-2,p}^{2(d+m+n),2\Delta_{1},d-1}{\frac{\left(\frac{1}{2}\left(2+d-2\Delta_{5}\right)\right)_{m-1}\left(\frac{1}{2}\left(2+d-2\Delta_{3}\right)\right)_{n-1}}{\left(1+{\frac{d}{2}}\right)_{m+n-1}}}\widehat{f}_{12,pnm}\,,$
(90)

where $\widehat{f}_{12,pnm}$ is universally solved by (79), while the matching
changes $c_{12,k_{1}k_{2}}$ to a much more compact expression
$c_{12g,k_{1}k_{2}}$

$\displaystyle c_{12g,k_{1}k_{2}}=c^{\rm
eff}_{56g}{\textstyle{\frac{\scriptstyle\left(d-2\Delta_{1}\right)\left(d-2\Delta_{3}\right)\Gamma(d-1)\Gamma\left(\Delta_{1}\right)\Gamma\left(\Delta_{3}\right)\left(\frac{2-d}{2}\right)_{k_{2}}\left(\frac{1}{2}\left(d+2k_{2}\right)\right)_{k_{1}}\Gamma\left(\frac{d}{2}-k_{2}\right)\Gamma\left(-\frac{d}{2}+k_{1}-k_{2}+1\right)}{\scriptstyle
32k_{1}!k_{2}!\left(d-2\Delta_{1}-2\Delta_{3}-2\right)\Gamma\left(\frac{d}{2}\right)\Gamma\left(\frac{d+2}{2}\right)\Gamma\left(\Delta_{1}+\Delta_{3}+1\right)\Gamma\left(-\frac{d}{2}-k_{2}+1\right)\Gamma\left(\frac{d}{2}+k_{1}-k_{2}\right)}}}\,.$
(91)

We find that our result can be rewritten in a format analogous to the scalar
Mellin Feynman rules Nandan:2011wc . We should slightly modify the vertex
function (see (135)) to count the number of derivatives

$\displaystyle V_{m_{1}\ldots
m_{k}}^{N,\Delta_{1}\ldots\Delta_{k}}=\sum_{n_{1}=0}^{m_{1}}\cdots\sum_{n_{k}=0}^{m_{k}}\Gamma\left(\frac{\sum_{j}\left(\Delta_{j}+2n_{j}\right)-d+N}{2}\right)\prod_{j=1}^{k}\frac{\left(-m_{j}\right)_{n_{j}}}{n_{j}!\left(\Delta_{j}-\frac{d}{2}+1\right)_{n_{j}}}.$
(92)

Then we verify that our result (90) and (91) can be rewritten by

$\displaystyle
f_{12,pnm}=-\frac{\left(\Delta_{1}-1\right)\left(\Delta_{3}-1\right)\left(\Delta_{5}-1\right)}{(d-2)^{3}\Gamma\left(-\frac{d}{2}+\Delta_{1}+\Delta_{3}+\Delta_{5}+2\right)}\times$
$\displaystyle
V_{00m}^{1,\Delta_{5}-1,\Delta_{5},d-2}V_{00n}^{1,\Delta_{3}-1,\Delta_{3},d-2}V_{00p}^{1,\Delta_{1}-1,\Delta_{1},d-2}V_{nmp}^{1,d-1,d-1,d-1}S_{m}^{d-2}S_{n}^{d-2}S_{p}^{d-2}\,.$
(93)

###  4.3 Other truncated six-point amplitudes

$5$$6$$1$$2$$3$$4$$1$$2$$3$$4$$5$$6$ Figure 10: Six-point Witten diagram with
gluon exchanges. The coil lines represent gluon, giving rise to single-trace
conserved current with $(\Delta=d-1,J=1)$.

For the Lagrangian (53) , we still have two diagrams for six-point amplitudes
as shown in Fig 10, see also Herderschee:2022ntr . In this subsection, we
apply our algorithm to efficiently calculate these two diagrams by focusing on
even integer dimensions and presenting results for identical scalars with
$\Delta=d/2+k$ for $k\in\mathbb{Z}$ so that the amplitudes are truncated. Our
target is to demonstrate the efficiency of our algorithm for computing
truncated amplitudes, thus we will not be too strict with the overall
normalization that can be absorbed into the YM coupling. We find patterns
analogous to scalar Mellin Feynman rules, which we believe the final
expressions are also valid for other dimensions and identical scalars.
Unfortunately, we are not able to conclude a general Mellin Feynman rules for
Witten diagrams involving gluons from our results.

The boundary Feynman rules can be easily constructed. More generally, since we
are not limited to massless scalar, we should slightly modify the boundary
Feynman rules constructed in Herderschee:2022ntr

$\displaystyle\widehat{\mathcal{D}}_{34}^{d-1,1}\widehat{\mathcal{D}}_{12}^{d-1,1}\circ
M_{12g,gg56,g34}=C_{gg}\big{(}\widehat{L}_{13}-\widehat{L}_{23}+\widehat{L}_{24}-\widehat{L}_{14}\big{)}\circ
1\,.$
$\displaystyle\widehat{\mathcal{D}}_{56}^{d-1,1}\widehat{\mathcal{D}}_{123}^{\Delta,0}\widehat{\mathcal{D}}_{12}^{d-1,1}\circ
M_{12g,g3s,s4g,g56}=C_{ggs}\big{(}\widehat{L}_{11}-\widehat{L}_{22}+2\widehat{L}_{13}-2\widehat{L}_{23}\big{)}\big{(}\widehat{L}_{55}-\widehat{L}_{66}+2\widehat{L}_{45}-2\widehat{L}_{46}\big{)}\circ
1\,,$ (94) 444Note that in general $\widehat{L}_{11}\neq\widehat{L}_{22}$, but
when acting on the correlation functions with $\Delta_{1}=\Delta_{2}$ we have
$\widehat{L}_{11}\circ 1=\widehat{L}_{22}\circ 1$, precisely embracing the
spirit of the on-shell condition for amplitudes.

where the color factors are given by

$\displaystyle C_{gg}=g_{\rm
YM}^{4}\,f^{12g_{1}}f^{34g_{2}}(f^{5b_{1}g_{2}}f^{6b_{1}g_{1}}+f^{5b_{1}g_{1}}f^{6b_{1}g_{2}})\,,\quad
C_{ggs}=g_{\rm YM}^{4}f^{12g_{1}}f^{23g_{1}}f^{24g_{3}}f^{56g_{3}}\,.$ (95)

Following our algorithm, we write the ansatz in a format as follows

$\displaystyle
M_{12g,gg56,g34}=P_{gg}\sum_{n=0,m=0}{\frac{f_{gg}^{mn}}{\left(\frac{d}{2}+\delta_{34}-\Delta+m-1\right)\left(\frac{d}{2}+\delta_{12}-\Delta+n-1\right)}}\,,$
$\displaystyle
M_{12g,g3s,s4g,g56}=P_{ggs}\sum_{m=0,n=0,p=0}{\textstyle{\frac{\scriptstyle
f_{ggs}^{mnp}}{\scriptstyle\left(\frac{d}{2}+\delta_{56}-\Delta+m-1\right)\left(\frac{d}{2}+\delta_{12}+\delta_{13}+\delta_{23}-\frac{3\Delta}{2}+n-1\right)\left(\frac{d}{2}+\delta_{12}-\Delta+p-1\right)}}}\,,$
(96)

where the kinematic factors are precisely given by the boundary Feynman rules

$\displaystyle
P_{gg}=2C_{gg}(6\Delta-d)(\delta_{13}-\delta_{14}-\delta_{23}+\delta_{24})\,,$
$\displaystyle P_{ggs}=16C_{ggs}(6\Delta-d)(6\Delta-d+2)\times$
$\displaystyle\left(\delta_{13}-\delta_{23}\right)\left(-2\delta_{12}-2\delta_{13}-\delta_{14}-2\delta_{15}-2\delta_{23}-\delta_{24}-2\delta_{25}-\delta_{34}-2\delta_{35}+3\Delta\right)\,.$
(97)

We are then ready for solving $f_{gg}^{nm}$ and $f_{ggs}^{mnp}$. We compute
these coefficients for $d=4,6,8,10$ and $\Delta=d/2+k$ for at most $k=5$. As
expected, we find a general pattern similar to the scalar Mellin Feynman
rules.

$\displaystyle
f_{gg}^{mn}={\textstyle{\frac{\scriptstyle(\Delta-1)^{2}}{\scriptstyle(d-2)^{2}\Gamma\big{(}3\Delta-{\frac{d}{2}}+1\big{)}}}}V_{00n}^{1;\Delta-1,\Delta,d-2}V_{00m}^{1;\Delta-1,\Delta,d-2}V_{00nm}^{0;\Delta,\Delta,d-1,d-1}S_{m}^{d-2}S_{n}^{d-2}\,,$
$\displaystyle
f_{ggs}^{mnp}=-{\textstyle{\frac{\scriptstyle(\Delta-1)^{2}}{\scriptstyle
2(d-2)^{2}\Gamma\big{(}3\Delta-{\frac{d}{2}}+2\big{)}}}}V_{00p}^{1;\Delta-1,\Delta,d-2}V_{00m}^{1;\Delta-1,\Delta,d-2}V_{0np}^{1;\Delta,\Delta,d-1}V_{0nm}^{1;\Delta,\Delta,d-1}S_{m}^{\Delta}S_{n}^{d-2}S_{p}^{d-2}\,,$
(98)

It is straightforward to check these results for higher dimensions and higher
$k$. We claim the results here should also be valid for general dimensions and
identical scaling dimensions $\Delta$.

###  4.4 Simple examples at eight-point

In this subsection, we provide solutions for two examples of eight-point
amplitudes involving gluon exchange. These examples are among the simplest
diagrams at the eight-point level. In particular, we will consider an example
that is not gauge invariant by itself, which requires a more cautious approach
when dealing with the Casimir cut.

####  4.4.1 Example I

In Fig 11, we consider an eight-point example, which is a comb diagram with
only two derivatives.

$1$$2$$3$$4$$5$$6$$7$$8$ Figure 11: The first example of eight-point Witten
diagrams with gluon exchanges.

The boundary Feynman rules for Fig 11 can be easily constructed as

$\displaystyle\widehat{\mathcal{D}}_{78}^{d-1,1}\widehat{\mathcal{D}}_{56}^{d-1,1}\widehat{\mathcal{D}}_{34}^{d-1,1}\widehat{\mathcal{D}}_{12}^{d-1,1}\circ
M_{12g,34g,56g,78g}$
$\displaystyle=C^{a}_{gggg}\left(\big{(}\widehat{L}_{15}-\widehat{L}_{16}+\widehat{L}_{26}-\widehat{L}_{25}\big{)}\big{(}\widehat{L}_{37}-\widehat{L}_{38}+\widehat{L}_{48}-\widehat{L}_{47}\big{)}-(5\leftrightarrow
7,6\leftrightarrow 8)\right)\circ 1\,$
$\displaystyle+C^{b}_{gggg}\left(\big{(}\widehat{L}_{13}-\widehat{L}_{14}+\widehat{L}_{24}-\widehat{L}_{23}\big{)}\big{(}\widehat{L}_{57}-\widehat{L}_{58}+\widehat{L}_{68}-\widehat{L}_{67}\big{)}-(3\leftrightarrow
7,4\leftrightarrow 8)\right)\circ 1\,$
$\displaystyle+C^{c}_{gggg}\left(\big{(}\widehat{L}_{13}-\widehat{L}_{14}+\widehat{L}_{24}-\widehat{L}_{23}\big{)}\big{(}\widehat{L}_{57}-\widehat{L}_{58}+\widehat{L}_{68}-\widehat{L}_{67}\big{)}-(3\leftrightarrow
5,4\leftrightarrow 6)\right)\circ 1.$ (99)

where $C^{a}_{gggg}$ encodes all the color factors and coupling constant and
are given by

$\begin{split}C^{a}_{gggg}&=g_{YM}^{6}f^{12a}f^{34b}f^{56c}f^{78d}f^{abe}f^{cde}\,,\quad
C^{b}_{gggg}=g_{YM}^{6}f^{12a}f^{34b}f^{56c}f^{78d}f^{ace}f^{bde}\,,\\\
C^{c}_{gggg}&=g_{YM}^{6}f^{12a}f^{34b}f^{56c}f^{78d}f^{ade}f^{bce}\,.\end{split}$

With the flat extension properties stated before, we can write down the
following simple ansatz

$\displaystyle
M_{12g,34g,56g,78g}=P_{gggg}\sum_{mnpq}{\textstyle{\frac{\scriptstyle
f^{mnpq}_{gggg}}{\scriptstyle(\frac{d}{2}+\delta_{12}-\Delta+m-1)(\frac{d}{2}+\delta_{34}-\Delta+n-1)(\frac{d}{2}+\delta_{56}-\Delta+p-1)(\frac{d}{2}+\delta_{78}-\Delta+q-1)}}}\,,$
(100)

where

$\begin{split}P_{gggg}=&4(8\Delta-d)^{2}\Big{[}C^{a}_{gggg}\left((\delta_{15}-\delta_{16}+\delta_{26}-\delta_{25})(\delta_{37}-\delta_{38}+\delta_{48}-\delta_{47})-(5\leftrightarrow
7,6\leftrightarrow 8)\right)\\\
&+C^{b}_{gggg}\left((\delta_{13}-\delta_{14}+\delta_{24}-\delta_{23})(\delta_{57}-\delta_{58}+\delta_{68}-\delta_{67})-(3\leftrightarrow
7,4\leftrightarrow 8)\right)\\\
&+C^{c}_{gggg}\left((\delta_{13}-\delta_{14}+\delta_{24}-\delta_{23})(\delta_{57}-\delta_{58}+\delta_{68}-\delta_{67})-(3\leftrightarrow
5,4\leftrightarrow 6)\right)\Big{]}\,.\end{split}$ (101)

We implemented the same strategy for solving the truncated case, and find the
following general pattern for $f^{mnpq_{gggg}}$

$\begin{split}f^{mnpq}_{gggg}=&\frac{(\Delta-1)^{4}}{(d-2)^{4}\Gamma(4\Delta-\frac{d}{2}+2)}V_{00m}^{1;\Delta-1,\Delta,d-2}V_{00n}^{1;\Delta-1,\Delta,d-2}V_{00p}^{1;\Delta-1,\Delta,d-2}V_{00q}^{1;\Delta-1,\Delta,d-2}\\\
&\times
V_{mnpq}^{0;d-1,d-1,d-1,d-1}S_{m}^{d-2}S_{n}^{d-2}S_{p}^{d-2}S_{q}^{d-2}\,.\end{split}$
(102)

####  4.4.2 Example II: cautions with nonphysical modes

For the second example, we consider the eight-point amplitudes as shown in Fig
12 below.

$\Pi_{56,78}$$1$$2$$3$$4$$5$$6$$7$$8$ Figure 12: The second example of eight-
point Witten diagrams with gluon exchanges, where the intermediate bulk-to-
bulk gluon propagator $\Pi_{56,78}$ carries nonphysical longitudinal modes.

This is an example that is not gauge invariant by itself. The culprit is the
bulk-to-bulk gluon propagator $\Pi_{56,78}$ connecting $\phi_{7}\phi_{8}$ pair
and $\phi_{5}\phi_{6}$ pair. To see this, note that the gluon bulk-to-bulk
propagators can be modified by adding the gauge-fixing term in the Lagrangian.
The gauge-fixing term will bring up spurious longitudinal degrees of freedom.
Nevertheless, the amplitudes, as gauge invariant objects, should be
independent of the gauge-fixing term. In previous examples, although we only
consider a single Witten diagram without summing over to final amplitudes, the
gluon bulk-to-bulk propagator always connects two external scalars anti-
symmetrically. This anti-symmetry and the on-shell conditions of external legs
then ensure the gauge invariance, as the fake longitudinal modes are
automatically vanishing by
$(L_{i}-L_{j})\cdot(L_{i}+L_{j})=L_{i}^{2}-L_{j}^{2}\equiv 0$. However, for
our example in Fig 12, the longitudinal modes in the propagator $\Pi_{56,78}$
survive because

$\displaystyle(L_{1}-L_{2})\cdot(L_{1}+L_{2}+L_{7}+L_{8})\neq 0\,.$ (103)

Surprisingly, our method knows that Fig 12 is not gauge-invariant by
attempting to solve the effective amplitudes $M^{\rm eff}_{1278g,3456g}$ in
Fig 13.

$\Pi_{56,78}$$1$$2$$3$$4$$5$$6$$7$$8$ Figure 13: The effective amplitude
$M^{\rm eff}_{1278g,3456g}$ after performing the Casimir cut
$\mathcal{C}_{12}$ and $\mathcal{C}_{34}$.

Naively, we can follow our routine to first make the ansatz

$\displaystyle M^{\rm
eff}_{1278g,3456g}=\sum_{m}{\frac{f_{1278g}^{m}P_{ggg}}{\frac{d}{2}+\delta_{12}+\delta_{17}+\delta_{18}+\delta_{27}+\delta_{28}+\delta_{78}-2\Delta+m-1}}\,,$
(104)

where

$\displaystyle
P_{ggg}=2(8\Delta-d)(\delta_{13}-\delta_{14}-\delta_{23}+\delta_{24})C_{ggg}\,,$
$\displaystyle C_{ggg}=-g_{\rm
YM}^{4}f^{12a_{1}}f^{34a_{2}}(f^{5ba_{3}}f^{6ba_{2}}+f^{5ba_{2}}f^{6ba_{3}})(f^{7ca_{3}}f^{8ca_{1}}+f^{7ca_{1}}f^{8ca_{3}}).$
(105)

Then we try to solve

$\displaystyle\widehat{\mathcal{D}}_{1278}^{d-1,1}\circ M^{\rm
eff}_{1278g,3456g}=C_{ggg}\big{(}\widehat{L}_{13}-\widehat{L}_{14}-\widehat{L}_{23}+\widehat{L}_{24}\big{)}\circ
1\,.$ (106)

However, we find that there is no solution for $f_{1278g}^{m}$. This issue is
anticipated, as the Witten diagram Fig 13 cannot be represented by the ansatz
(104). Instead, (104) only captures the transverse part of Fig 13. Then the
question is that how can we obtain the transverse part of the amplitude in Fig
12 using our algorithm without knowing its gauge-invariant completion?

Our trick is to append the possible longitudinal contribution $M_{L}$, and the
correct Casimir equation is

$\displaystyle\widehat{\mathcal{D}}_{1278}^{d-1,1}\circ M^{\rm
eff}_{1278g,3456g}+M_{L}=C_{ggg}\big{(}\widehat{L}_{13}-\widehat{L}_{14}-\widehat{L}_{23}+\widehat{L}_{24}\big{)}\circ
1\,.$ (107)

To proceed, we should make an ansatz for $M_{L}$. It is worth noting that
$M_{L}$ is not contact, rather it contains all the poles that $M^{\rm
eff}_{1278g,3456g}$ have (and probably more). An intuitive way to understand
this is to think about its flat-space cousin: the longitudinal mode
contributes an additional $1/p^{2}$. Formally speaking, the longitudinal part
of the propagator (after the Casimir cut) gives

$\displaystyle\nabla_{\mu}\nabla^{\rho}(\Pi_{56,78})_{\rho\nu}\,.$ (108)

As the derivatives are moved to the external legs by the integration-by-parts,
we are still left with bulk-to-bulk propagators that contribute poles in
Mellin variables. Besides, during this procedure, we have a new vertex
$V_{1278}$, which can help us make an appropriate ansatz for $M_{L}$.
Ultimately, we find that similar to the flat-space, the longitudinal modes
should be proportional to

$\displaystyle(L_{1}-L_{2})\cdot(L_{1}+L_{2}+L_{7}+L_{8})\times(L_{3}-L_{4})\cdot(L_{1}+L_{2}+L_{7}+L_{8})\,,$
(109)

motivating an ansatz

$\displaystyle
M_{L}=\sum_{m}{\frac{f_{L}^{m}(\delta_{27}+\delta_{28}-\delta_{17}-\delta_{18})(\delta_{45}+\delta_{46}-\delta_{35}-\delta_{36})}{\frac{d}{2}+\delta_{12}+\delta_{17}+\delta_{18}+\delta_{27}+\delta_{28}+\delta_{78}-2\Delta+m-1}}\,,$
(110)

This ansatz allows us to solve (107) and obtain the results for Fig 12 with
the only transverse modes

$\displaystyle
M_{12g,g78,g56,g34}^{T}=P_{ggg}\sum_{mnp}{\textstyle{\frac{\scriptstyle
f_{ggg}^{mnp}}{\scriptstyle({\frac{d}{2}}+\delta_{12}-\Delta+m-1)({\frac{d}{2}}+\delta_{34}-\Delta+n-1)(\frac{d}{2}+\delta_{12}+\delta_{17}+\delta_{18}+\delta_{27}+\delta_{28}+\delta_{78}-2\Delta+p-1)}}}\,,$
(111)

where

$\displaystyle f_{ggg}^{mnp}={\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2(d-2)^{2}\Gamma\big{(}1+4\Delta-{\frac{d}{2}}\big{)}}}}V_{00m}^{1;\Delta-1,\Delta,d-2}V_{00n}^{1;\Delta-1,\Delta,d-2}V_{00pn}^{0;\Delta,\Delta,d-1,d-1}V_{00pm}^{0;\Delta,\Delta,d-1,d-1}S_{m}^{d-2}S_{n}^{d-2}S_{p}^{d-1}\,.$
(112)

##  5 Further remarks

###  5.1 Relation to the cosmological correlators

In this paper, we focus on the amplitudes in AdS. Nevertheless, it turns out
that the late-time correlators in de Sitter (dS) space, are closely related to
AdS amplitudes through analytic continuation Sleight:2021plv ; Sleight:2020obc
; DiPietro:2021sjt . Typically, transitioning from the Euclidean AdS
background to dS is rather straightforward; one can simply Wick rotate the AdS
radial coordinates to become the dS time, as well as Wick rotate the AdS
radius to the Hubble radius. For example, in the Poincaré patch, we have:

$\displaystyle ds_{\rm AdS}=\frac{R_{\rm
AdS}^{2}}{z^{2}}(dz^{2}+dx^{2})\underset{z\rightarrow-it}{\overset{R_{\rm
AdS}\rightarrow-iH_{\rm dS}}{\mkern
4.5mu\relbar\mkern-4.5mu\xrightarrow{\hskip 56.9055pt}}}ds_{\rm
dS}=\frac{H_{\rm dS}^{2}}{t^{2}}(-dt^{2}+dx^{2}),.$ (113)

As is always the case in quantum field theories, the corresponding analytic
continuation of analytic functions, such as scattering amplitudes, is highly
nontrivial eden2002analytic . We refer readers to Sleight:2021plv ;
Sleight:2020obc  for details on analytically continuing the AdS amplitudes to
describe the cosmological correlator.

In this section, however, we claim that the Casimir equations for any Witten
diagrams remain valid and intact after the analytic continuation (except for
switching the signs of those parameters that explicitly depend on the
cosmological constant). The reason is that the Casimir equations consist of
the boundary conformal generators that are purely defined on either the AdS
boundary or the late-time surface in dS, regardless of the analytic properties
of the functions these equations solve. Indeed, if we look into the
differential representation of four-point functions as in Fig 3 (but with
$J=0$), generally we have the Casimir cut

$\displaystyle\mathcal{D}^{\Delta,0}_{12}M_{4}={\rm contact}\,,$ (114)

where we should emphasize that $\mathcal{D}_{12}^{\Delta,0}$ is invariant
under the shadow $\Delta\rightarrow d-\Delta$. This equation, as Fourier
transform to the momentum space, is precisely the cosmological bootstrap
equation proposed in Arkani-Hamed:2018kmz  (specifically see eq (2.41) for
$\phi^{3}$ theory and eq (2.46) for a more general statement in Arkani-
Hamed:2018kmz )! Our exploration in this paper is then expected to provide
solid and well-defined cosmological bootstrap equations for higher-point
correlators, which may benefit understanding higher spectrum in the slow-roll
scenario of the inflation Arkani-Hamed:2018kmz ; Baumann:2022jpr .

Nonetheless, for the same equation, how do we know whether we are solving the
AdS amplitudes or cosmological correlators? This is actually reminiscent of
the differential equation method for solving loop Feynman diagrams
Kotikov:1990kg ; Remiddi:1997ny ; Gehrmann:1999as ; Argeri:2007up ;
Henn:2014qga , where the Feynman integrals from any analytic contour (e.g.,
under certain unitary cuts) share the same differential equations. Different
contours, as reflected by different analytic properties of the integral
results, can be accounted for by the boundary conditions of the differential
equations Anastasiou:2002qz ; Anastasiou:2002yz ; Henn:2014qga . (A)dS tree-
level correlators follow a similar pattern: we integrate out the bulk
coordinates to obtain the amplitudes. The analytic continuation is performed
for the bulk coordinates only; therefore, we may find appropriate bulk
integral contour in AdS and then deform this contour to achieve the transition
from AdS to dS Sleight:2019hfp ; Sleight:2019mgd ; Sleight:2021plv 555Another
well-known example is to define the conformal blocks by the geodesic Witten
diagram Hijano:2015zsa ; Hijano:2015qja , where the integral contour is placed
to align with the bulk geodesics.. From the perspective of the Casimir
equations, we simply pick up an appropriate boundary condition, as Arkani-
Hamed:2018kmz  did in momentum space. This is a very standard trick when it
comes to the analytic continuation of analytic functions, a well-known and
most relevant example is the Casimir equation for the conformal block (47):
there are $8$ independent solutions to this Casimir equation with a given
$(\Delta,J)$, and they have different asymptotic behaviours; the analytic
continuation then picks up specific combinations of these solutions Caron-
Huot:2017vep ; Kravchuk:2018htv . The Casimir equations for (A)dS correlators
should follow similarly, for example, $\Delta_{+}=\Delta$ mixes with its
shadow $\Delta_{-}=d-\Delta$ in dS correlators.

However, this argument may raise a puzzle for our algorithm: solutions of
Casimir equations are ambiguous, as there are integration constants to be
determined by the boundary conditions; however, we find no such constants in
the Mellin space. The explanation is that we have already fixed the relevant
boundary conditions by making the Mellin ansatz. Our ansatz of Mellin
amplitudes is made by the OPE analysis in the unitary CFT, which ensures that
we are solving the AdS amplitudes rather than the dS ones. Therefore, a
shortcut is that we may be able to make appropriate ansatz for the
cosmological correlators for either momentum space Arkani-Hamed:2018kmz ;
Baumann:2022jpr , Mellin space Penedones:2010ue ; Sleight:2020obc  or
momentum-Mellin space Sleight:2019hfp ; Sleight:2019mgd  and solve the
differential representation unambiguously. For example, it turns out that the
correct Mellin ansatz for the four-point cosmological correlators (defined in
the Bunch-Davis vacuum) is Sleight:2020obc

$\displaystyle M^{\rm
dS}_{\nu,J}(s,t)={\frac{1}{2}}\sin\big{(}{\frac{\pi}{2}}(\Delta_{\Sigma}+2J-d)\big{)}M^{\rm
AdS}_{\Delta_{+},J}(s,t)$
$\displaystyle-\sin\big{(}{\frac{\pi}{2}}(\Delta_{1}+\Delta_{2}+\Delta_{-}+J-d)\big{)}\sin\big{(}{\frac{\pi}{2}}(\Delta_{3}+\Delta_{4}+\Delta_{-}+J-d)\big{)}\mathcal{F}_{\nu,J}^{\rm
AdS}(s,t)\,,$ (115)

where $\Delta_{\pm}=d/2\pm i\nu$ and $\mathcal{F}_{\nu,J}^{\rm AdS}$ is the
Mellin transform of the conformal partial waves. In the future, it would be of
great interest to understand the cosmological ansatz for any point and then
generalize our method to efficiently compute the cosmological correlators to
higher points.

###  5.2 Flat-space limit

In the end, we would like to comment on the flat-space limit of the
perturbative AdS scattering. This is a kinematic limit where the AdS radius is
much larger than any other scales, e.g.,

$\displaystyle R_{\rm AdS}\gg b>\ell_{s}\gg\ell_{pl}\,,$ (116)

where $b$ is the impact parameter and $\ell_{s}$ is the string scale.
Intuitively the flat-space limit of AdS observables should give us back their
flat-space versions. However, it is generally a hard task to verify and prove
such a statement. There are different formalisms for extracting the flat-space
observables from AdS amplitudes Raju:2012zr ; Penedones:2010ue ;
Fitzpatrick:2011hu ; Paulos:2016fap ; Okuda:2010ym ; Maldacena:2015iua ;
Komatsu:2020sag ; Hijano:2019qmi ; Hijano:2020szl ; Caron-Huot:2021kjy ;
Cordova:2022pbl ; Jain:2022ujj , see also Li:2021snj  for connections between
these formalisms, although the underlying analyticity remains subtle
Komatsu:2020sag ; vanRees:2022itk  (for extension to supersymmetric
amplitudes, see, e.g., Alday:2017vkk ; Rastelli:2016nze ; Rastelli:2017udc ;
Alday:2019nin ; Alday:2020dtb ; Alday:2021odx ). Surprisingly, the flat-space
scattering amplitudes can reversely shed light on the local AdS physics at the
AdS Regge limit Cornalba:2006xk ; Antunes:2020pof ; Caron-Huot:2021enk

$\displaystyle R_{\rm AdS}\sim b\gg\ell_{s}\gg\ell_{pl}\,.$ (117)

In this limit, flat-space causality constraints can be lifted to AdS with
small errors Caron-Huot:2021enk .

In this subsection, we emphasize that the differential representation is even
more powerful, as it completely uplifts (at least tree-level) the flat-space
amplitudes to AdS schematically by replacing the momenta with the conformal
generators. For spinning particles, it is also necessary to include the
weight-shifting operators Karateev:2017jgd , see Li:2022tby ; Lee:2022fgr  for
discussions. See also Bissi:2022wuh  for the differential representation for
supersymmetric amplitudes.666As the differential representation uplifts flat-
space amplitudes, it is also expected to provide a formal explanation for the
double-copy structures of supersymmetric amplitudes in AdS observed by
Zhou:2021gnu . Therefore, the flat-space limit is a corner of this uplifting
map, which is now made manifest by the differential representation. The
differential representation then provides a more comprehensive way to
understand the reason that the flat-space limit should work. We can see this
by considering the Inönü-Wigner contraction of the conformal generators
Giddings:1999jq ; Goncalves:2014rfa

$\displaystyle p_{\mu}=\lim_{R_{\rm AdS}\rightarrow\infty}{\frac{1}{R_{\rm
AdS}}}L_{0\mu}\,,\quad s_{\mu\nu}=L_{\mu\nu}\,,$ (118)

where $\mu,\nu$ are indices of the flat-space $\mathbb{R}^{d,1}$; $p_{\mu}$
and $s_{\mu\nu}$ are the translation and the rotation (with the boost) in
$\mathbb{R}^{d,1}$. Therefore, the Inönü-Wigner contraction of the conformal
generators gives rise to the Poincare generators in the flat-space
Giddings:1999jq ; Goncalves:2014rfa . Similarly, the Inönü-Wigner of the
quadratic conformal Casimir gives rise to the momentum squared in the flat-
space

$\displaystyle(\sum_{i=1}^{n}p_{i})^{2}\sim\lim_{R_{\rm
AdS}\rightarrow\infty}{\frac{1}{R_{\rm AdS}^{2}}}\mathcal{C}_{1\cdots n}\,.$
(119)

It is obvious that this provides a group theoretical understanding of the
flat-space limit from the differential representation, provided the fact that
the $\phi^{n}$ contact Witten diagram in the flat-space limit is reduced to
the momentum conservation delta function Komatsu:2020sag ; Li:2021snj ,
because

$\displaystyle R_{\rm AdS}^{-2}\,L_{i}\cdot L_{j}\rightarrow p_{i}\cdot
p_{j}+\mathcal{O}(R_{\rm AdS}^{-2})\,.$ (120)

This may provide a systematic way to expand in $1/R_{\rm AdS}$ as a probe to
study the Infra-Red (IR) divergence of flat-space S-matrix by viewing $R_{\rm
AdS}$ as IR cut-off, e.g., see Banerjee:2022oll ; Duary:2022pyv ;
Duary:2022afn . We can also easily understand the flat-space limit in the
Mellin space Penedones:2010ue ; Paulos:2016fap : $L_{i}\cdot L_{j}$ acting on
the Mellin amplitudes shifts the Mellin variables, at the large scaling limit
of the Mellin variables, the action is dominated by multiplying $\delta_{ij}$
to the Mellin amplitudes. We can easily verify all our results in this paper
can be reduced to the correct flat-space amplitudes by scaling the Mellin
variables $\delta_{ij}\rightarrow\infty$.

##  6 Summary

In this paper, we have proposed a more efficient approach for computing Witten
diagrams in AdS using differential representation. Our investigations
emphasized that the differential representation is not merely a schematic
representation of the amplitudes but should be interpreted as the Casimir
equations of Witten diagrams. We used Casimir cuts to visualize the Casimir
equations, removing the targeted bulk-to-bulk propagators which provide an
iteration to build any tree-level Witten diagram from contact Witten diagrams
that are systematically constructed by the boundary Feynman rules. As
expected, these boundary Feynman rules directly uplift the flat-space Feynman
rules (without higher derivative interactions).

We then explored how we can solve the Casimir equations of Witten diagrams
iteratively and efficiently. Our strategy is to study the Witten diagrams in
Mellin space, where the differential equations from the Casimir cuts become
difference equations in terms of the Mellin variables. In this case, we
proposed an algorithm that allows us to bootstrap Witten diagrams in Mellin
space from OPE analysis (building the ansatz), analyticity (the cancellation
of pole structures), and consistency (matching to the boundary Feynman rules).
However, it is worth noting that the term “bootstrap” used here differs from
the notion of the nonperturbative bootstrap because we are literally doing
perturbative calculations and need the Lagrangian to build the diagrams and
computing rules. Nevertheless, our method still embodies the spirit of
bootstrap by by utilizing symmetry, analyticity, and consistency to obtain
results. Using our algorithm, we easily reproduced the four-point Witten
diagrams from scalar, gluon, and graviton exchange. Notably, the graviton
exchange diagram required efforts to address the contact source terms using
standard techniques Costa:2014kfa , while our method is more straightforward
and efficient.

As for new results, we adapted our algorithm for computing six-point scalar
Witten diagrams, where we allowed both the gluon and scalar to be mediated. We
found that these amplitudes take a form similar to scalar Mellin Feynman rules
Penedones:2016voo , but with modified vertex and propagating Gamma functions.
Unfortunately, we have not found clean Mellin Feynman rules involving gluons,
due to the limited examples. Additionally, we tested our algorithm by solving
two examples of eight-point amplitudes involving gluon exchange and
demonstrated how to obtain the transverse part of a single non-gauge-invariant
Witten diagram.

In the end, we provided non-technical comments on how our methods could be
potentially generalized to bootstrap the cosmological correlators in dS, and a
simple remark on manifesting the flat-space limit using the differential
representation. It is of great interest to follow our routine and compute the
cosmological correlators in the future. Besides, it will also be crucial to
understand the subtlety of the analyticity in the flat-space limit by using
the differential representation.

There are several other important areas for future exploration. One natural
extension of our work is to generalize our methods to compute AdS amplitudes
at the loop level using the differential representation of loop Witten
diagrams Aharony:2016dwx ; Herderschee:2021jbi . Furthermore, it is crucial to
extend our method to compute Witten diagrams involving external spinning
particles, such as gluons and gravitons Li:2022tby . This investigation will
pave the way for understanding the renormalization flow in AdS and its CFT
dual Bertan:2018afl ; Aharony:2016dwx ; Aprile:2017bgs ; Bertan:2018khc ;
Antunes:2021abs , and may facilitate the computation of loop observables in
AdS, e.g.,Yuan:2017vgp ; Yuan:2018qva ; Huang:2021xws ; Huang:2023oxf . In
addition, discovering the hidden structure among the gauge theory, such as the
BCJ relation and the double copy relation Bern:2019prr ; Bern:2008qj ;
Bern:2010ue , would be important for future research. Recent developments in
curved space can be found in Armstrong:2020woi ; Farrow:2018yni ;
Lipstein:2019mpu ; Albayrak:2020fyp ; Alday:2021odx ; Zhou:2021gnu ;
Bissi:2022wuh ; Armstrong:2023phb ; Lipstein:2023pih . Furthermore, our method
can likely be applied to supergravity by incorporating the AdS version of SUSY
delta functions. This implies a brute force method (other than conformal
bootstrap Gopakumar:2022kof ) to compute the Virasoro-Shapiro amplitude in
${\rm AdS}_{5}\times{\rm S}_{5}$ (see Abl:2020dbx ; Aprile:2020mus ;
Alday:2022xwz ; Alday:2022uxp ): expand the Kaluza-Klein reduction and
evaluate the diagram term by term. Finally, our results also provide important
ingredients for performing the Mellin bootstrap program Gopakumar:2016wkt ;
Gopakumar:2016cpb  for higher-point correlation functions in holographic CFTs,
which could shed light on understanding the multi-twist operators
Antunes:2021kmm .

Acknowledgements

We thank Agnese Bissi, Simon Caron-Huot, Arthur Lipstein, Julio Parra-
Martinez, David Simmons-Duffin for useful discussions. We are also grateful to
Fei Teng and Xinan Zhou for valuable comments on the draft. YZL is supported
by the Simons Foundation through the Simons Collaboration on the
Nonperturbative Bootstrap. JM is supported by a Durham-CSC Scholarship.

##  Appendix A Boundary Feynman rules

###  A.1 Boundary Feynman rules by exchanging gravitons

We take graviton as example to show how we construct the boundary Feynman
rules by addressing the trace of gauge fields. We follow the logic of
Li:2022tby  to present. It turns out that it is instructive to decompose the
graviton into traceless part and trace part (with de-Donder gauge) at the
level of the Lagrangian.

$\displaystyle
h_{\mu\nu}=\tilde{h}_{\mu\nu}+{\frac{1}{d+1}}hg_{\mu\nu}\,,\quad\tilde{h}\equiv
0\,.$ (121)

We can then treat $\tilde{h}_{\mu\nu}$ and $h$ separately. In the embedding
formalism, they give rise to two different bulk-to-bulk propagators by the
wick contractions

$\displaystyle
W_{1}^{\mu}W_{1}^{\nu}W_{2}^{\rho}W_{2}^{\sigma}\left\langle\tilde{h}_{\mu\nu}(Y_{1})\tilde{h}_{\rho\sigma}(Y_{2})\right\rangle=\Pi_{bb}^{d,2}(Y_{1},Y_{2};W_{1},W_{2})\,,$
$\displaystyle\left\langle h(Y_{1})h(Y_{2})\right\rangle=({\rm
Tr}\,\Pi)_{bb}^{d,2}(Y_{1},Y_{2})\,.$ (122)

In the second line, we do not write $W$ dependence to explicitly remind us
that there are no polarizations in this case. Typically, when we perform this
decomposition, any vertex function coupled to the stress-tensor also
effectively makes this decomposition

$\displaystyle
V_{\mu\nu}h^{\mu\nu}=\tilde{V}_{\mu\nu}\tilde{h}^{\mu\nu}+{\frac{1}{d+1}}{\rm
Tr}\,V\,h\,.$ (123)

Without losing the generality, we consider $2$-to-$n$ scalar scattering by
exchanging graviton with effective vertices $V^{12}_{\mu\nu}$ and $V^{1\cdots
n}_{\mu\nu}$

$\displaystyle W_{2\rightarrow n}=\int
D^{d+2}Y_{1}D^{d+2}Y_{2}\,\Big{(}\tilde{V}^{12}_{\mu\nu}\left\langle\tilde{h}^{\mu\nu}(Y_{1})\tilde{h}^{\rho\sigma}(Y_{2})\right\rangle\tilde{V}^{1\cdots
n}_{\rho\sigma}+{\frac{1}{(d+1)^{2}}}{\rm Tr}\,V^{12}\left\langle
h(Y_{1})h(Y_{2})\right\rangle{\rm Tr}\,V^{1\cdots n}$
$\displaystyle+{\frac{1}{d+1}}{\rm Tr}\,V^{12}\left\langle
h(Y_{1})\tilde{h}^{\rho\sigma}(Y_{2})\right\rangle\tilde{V}^{1\cdots
n}_{\rho\sigma}+{\frac{1}{d+1}}\tilde{V}^{12}_{\mu\nu}\left\langle\tilde{h}^{\mu\nu}(Y_{1})h(Y_{2})\right\rangle{\rm
Tr}\,V^{1\cdots n}\Big{)}\,.$ (124)

Our goal is to show how we can construct the boundary Feynman rules by
following the flat-space routine. As we show in subsection 2.2 by using the
embedding formalism, we have

$\displaystyle\mathcal{D}_{12}W_{2\rightarrow n}=\int
D^{d+2}Y_{1}D^{d+2}Y_{2}\times$
$\displaystyle\Big{(}\tilde{V}_{\mu\nu}^{12}(\nabla_{Y_{1}}^{2}+2)\left\langle\tilde{h}^{\mu\nu}(Y_{1})\tilde{h}^{\rho\sigma}(Y_{2})\right\rangle\tilde{V}^{1\cdots
n}_{\rho\sigma}+{\frac{1}{(d+1)^{2}}}{\rm
Tr}\,V^{12}(\nabla_{Y_{1}}^{2}-2d)\left\langle
h(Y_{1})h(Y_{2})\right\rangle{\rm Tr}\,V^{1\cdots n}$
$\displaystyle+{\frac{1}{d+1}}{\rm
Tr}\,V^{12}(\nabla_{Y_{1}}^{2}-2d)\left\langle
h(Y_{1})\tilde{h}^{\rho\sigma}(Y_{2})\right\rangle\tilde{V}^{1\cdots
n}_{\rho\sigma}+{\frac{1}{d+1}}\tilde{V}^{12}_{\mu\nu}(\nabla_{Y_{1}}^{2}+2)\left\langle\tilde{h}^{\mu\nu}(Y_{1})h(Y_{2})\right\rangle{\rm
Tr}\,V^{1\cdots n}\Big{)}\,.$ (125)

In the next, we have to use the equation of motion in the de-Donder gauge

$\displaystyle(\nabla_{Y_{1}}^{2}+2)\left\langle
h_{\mu\nu}(Y_{1})h_{\rho\sigma}(Y_{2})\right\rangle-2g_{\mu\nu}\left\langle
h(Y_{1})h_{\rho\sigma}(Y_{2})\right\rangle={\frac{1}{2}}\big{(}g_{\mu\rho}g_{\nu\sigma}+g_{\mu\sigma}g_{\nu\rho}-{\frac{2g_{\mu\nu}g_{\rho\sigma}}{d-1}}\big{)}\delta(Y_{1}-Y_{2})\,.$
(126)

It is easy to find that upon the trace decomposition (125) we have

$\displaystyle(\nabla_{Y_{1}}^{2}+2)\left\langle\tilde{h}_{\mu\nu}(Y_{1})\tilde{h}_{\rho\sigma}(Y_{2})\right\rangle={\frac{1}{2}}\big{(}g_{\mu\rho}g_{\nu\sigma}+g_{\mu\sigma}g_{\nu\rho}-{\frac{2g_{\mu\nu}g_{\rho\sigma}}{d+1}}\big{)}\delta(Y_{1}-Y_{2})\,,$
$\displaystyle(\nabla_{Y_{1}}^{2}-2d)\left\langle
h(Y_{1})h(Y_{2})\right\rangle={\frac{2(d+1)}{d-1}}\delta(Y_{1}-Y_{2})\,,$
$\displaystyle(\nabla_{Y_{1}}^{2}-2d)\left\langle
h(Y_{1})\tilde{h}_{\mu\nu}(Y_{2})\right\rangle=(\nabla_{Y_{1}}^{2}+2)\left\langle
h_{\mu\nu}(Y_{1})h(Y_{2})\right\rangle=0\,,$ (127)

where the first two lines are equivalent to (19) and (21) for massless
spin-$2$ field. Plugging back to (125), it is then straightforward to find

$\displaystyle\mathcal{D}_{12}W_{2\rightarrow n}=\int
D^{d+2}Y_{1}\,V_{\mu\nu}^{12}\times{\frac{1}{2}}\big{(}g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho}-{\frac{2g^{\mu\nu}g^{\rho\sigma}}{d-1}}\big{)}\times
V_{\rho\sigma}^{1\cdots n}\,,$ (128)

namely, the propagator tensor is precisely the same as in flat-space.

###  A.2 Summarizing the boundary Feynman rules

In this section, we summarize the boundary Feynman rules for future reference.
Sticking to the Lorentz gauge, interaction vertex and the Casimir cut of the
gluon bulk-to-bulk propagator are given by

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}
\par{{}}{}{{}}{}{{}}\pgfsys@moveto{-28.45276pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-31.4655pt}{5.47743pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mu$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.98285pt}{3.533pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\nu$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \par
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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\@add@centering\end{minipage}=g^{\mu\nu}\vspace{10pt}\@add@centering$ (129)
$\text{The conserved current:}\centering\begin{minipage}{195.12767pt}
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{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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{}{}{}\pgfsys@moveto{-19.91684pt}{-19.91684pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{-19.91684pt}{19.91684pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{} {}{}{} {}{}{} {}{}{} }{{{{}{}{{}} }}{{}} {}{}{} {}{}{} {}{}{} {}{}{}
}{{{{}{}{{}} }}{{}} {}{}{} {}{}{} {}{}{} {}{}{} }{{{{}{}{{}} }}{{}} {}{}{}
{}{}{} {}{}{} {}{}{} }{{{{}{}{{}} }}{{}} {}{}{} {}{}{} {}{}{} {}{}{}
}{{{{}{}{{}} }}{{}} {}{}{} {}{}{} }{{{{}{}{{}} }}{{}} {{}} }
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}
\par{{}}{}{{}}{}{{}}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.44002pt}{5.47743pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mu$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \par
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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\@add@centering\end{minipage}=if^{abc}g_{YM}(\phi_{j}\nabla_{\mu}\phi_{i}-\phi_{i}\nabla_{\mu}\phi_{j})\vspace{10pt}\@add@centering$
(130)

Nest let’s consider gravity, for which we take the de-Donder gauge. As we
explained in the previous subsection, the corresponding Feynman rules are

$\hskip 5.69046pt\text{Graviton propagator
cut:}\centering\begin{minipage}{195.12767pt} \centering \leavevmode\hbox
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}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
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{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}}
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}
\par{{}}{}{{}}{}{{}}\pgfsys@moveto{-28.45276pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-33.93541pt}{5.47743pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mu\nu$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{23.01062pt}{5.47743pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\rho\sigma$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \par
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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\@add@centering\end{minipage}={\frac{1}{2}}\big{(}g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho}-{\frac{2g^{\mu\nu}g^{\rho\sigma}}{d-1}}\big{)}\vspace{10pt}\@add@centering$
(131)

$\displaystyle\text{The stress tensor:}\centering\begin{minipage}{195.12767pt}
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{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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{}{}{}\pgfsys@moveto{-19.91684pt}{-19.91684pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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$\displaystyle(\nabla_{\mu}\phi_{i}\nabla_{\nu}\phi_{j}+\nabla_{\nu}\phi_{i}\nabla_{\mu}\phi_{j}$
(132) $\displaystyle-
g_{\mu\nu}(\nabla_{\rho}\phi_{i}\nabla^{\rho}\phi_{j}+\Delta_{i}(\Delta_{i}-d)\phi_{i}\phi_{j}))\vspace{10pt}$

To obtain the final boundary Feynman rules, we need (28) to rewrite the bulk
derivative as boundary differential operators, therefore arriving at the
Casimir cut equations for the corresponding Witten diagram. However, we should
emphasize that the Casimir cut of the propagator we record here only
constructs the correct boundary Feynman rules for the gauge-invariant
diagrams, because we literally have fixed the gauge choice. For those Witten
diagrams with explicit gauge dependence by themselves, we refer the subsection
4.4.2 as a prototype for the cautious treatment.

##  Appendix B Scalar Mellin Feynman rules

In this section, we provide a brief review of the scalar Mellin Feynman rules,
which were summarized in Penedones:2016voo . These rules provide a powerful
method for calculating Witten diagrams in Mellin space involving scalar fields
with zero derivatives.

To apply the Mellin Feynman rules, we first assign a fictitious momentum
$p_{j}$ to every line in the Witten diagram, satisfying the on-shell condition
$-p_{i}^{2}=\Delta_{i}$ for external lines. Momentum is conserved at every
vertex of the diagram.

For each internal line, the propagator has a factor given by

$\frac{S_{m_{j}}^{\Delta_{j}}}{p_{j}^{2}+\Delta_{j}+2m_{j}},$ (133)

where $\Delta_{j}$ is the dimension of the propagating scalar field, and for
internal momentum $p_{ij}=\delta_{ij}$ the corresponding Mellin variables.

The propagator numerator is given by

$S_{m}^{\Delta}=\frac{\Gamma(\Delta-\frac{d}{2}+1+m)}{2(m!)\Gamma^{2}(\Delta-\frac{d}{2}+1)}.$
(134)

Finally, the vertex function is given by

$V_{m_{1}\ldots
m_{k}}^{\Delta_{1}\ldots\Delta_{k}}=\sum_{n_{1}=0}^{m_{1}}\cdots\sum_{n_{k}=0}^{m_{k}}\Gamma\left(\frac{\sum_{j}(\Delta_{j}+2n_{j})-d}{2}\right)\prod_{j=1}^{k}\frac{(-m_{j}){n_{j}}}{n_{j}!(\Delta_{j}-\frac{d}{2}+1){n_{j}}}.$
(135)

Lastly, we sum over all integers $m_{j}$ from the internal lines, and each sum
runs from 0 to $\infty$.

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