††institutetext: Department of Physics, University of Chicago, Chicago, IL
60637, USA

# Towards the Feynman rule for $n$-point gluon Mellin amplitudes in AdS/CFT

Jinwei Chu  Savan Kharel<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate the embedding formalism in conjunction with the Mellin
transform to determine tree-level gluon amplitudes in AdS/CFT. Detailed
computations of three to five-point correlators are conducted, ultimately
distilling what were previously complex results for five-point correlators
into a more succinct and comprehensible form. We then proceed to derive a
recursion relation applicable to a specific class of $n$-point gluon
amplitudes. This relation is instrumental in systematically constructing
amplitudes for a range of topologies. We illustrate its efficacy by
specifically computing six to eight-point functions. Despite the complexity
encountered in the intermediate steps of the recursion, the higher-point
correlator is succinctly expressed as a polynomial in boundary coordinates,
upon which a specific differential operator acts. Remarkably, we observe that
these amplitudes strikingly mirror their counterparts in flat space,
traditionally computed using standard Feynman rules. This intriguing
similarity has led us to propose a novel dictionary: comprehensive rules that
bridge AdS Mellin amplitudes with flat-space gluon amplitudes.

##  1 Introduction

In recent decades, the study of holographic theories has become a significant
area of theoretical research. Among these theories, the most developed are
those within the framework of asymptotically Anti-de Sitter (AdS) spacetimes
Maldacena:1997re ; Witten:1998qj . These theories present a contrast to the
traditional ‘in’ and ‘out’ states found in Minkowski spacetime, crucial for
scattering amplitudes. In AdS spacetimes, particles are intrinsically
confined, leading to perpetual interactions. Despite this, interactions at the
timelike boundary of AdS permit the creation and annihilation of particles
within this spacetime. Notably, these transition amplitudes in AdS have a
direct analogy to correlation functions in the corresponding Conformal Field
Theory (CFT). This correlation allows for the interpretation of CFT
correlation functions as scattering amplitudes in the AdS context.

Scattering amplitudes in Anti-de Sitter (AdS) space can be computed using
Witten diagrams, which are the AdS counterparts of Feynman diagrams used in
flat space. Initial efforts by researchers to extend their analysis beyond
three or four-point Witten diagrams faced significant computational challenges
(e.g. Liu:1998ty ; Freedman:1998bj ; DHoker:1998bqu ; Liu:1998th ;
DHoker:1998ecp ). Two primary difficulties emerged in this field: the
complexity of bulk integrals and the intricacies involved in dealing with
spin-bearing external operators. Overcoming these challenges has defined much
of the ongoing research in this area. In this paper, we directly address both
these challenges. We will embark on the calculation of higher-point external
spinning field.

Our work draws inspiration from the recent wave of diverse and intriguing
contributions to the computation of both scalar and spinning correlators in
AdS, employing varied methodologies: momentum space Raju:2012zr ;
Albayrak:2018tam ; Albayrak:2019yve ; Albayrak:2019asr ; Albayrak:2020bso ;
Albayrak:2020isk ; Albayrak:2020fyp ; Albayrak:2023kfk ; Bzowski:2013sza ;
Bzowski:2015pba ; Bzowski:2018fql ; Bzowski:2019kwd ; Bzowski:2020kfw ;
Bzowski:2022rlz ; Bzowski:2023jwt ; Jain:2020rmw ; Jain:2021vrv ; Jain:2023idr
; Isono:2018rrb ; Isono:2019wex ; Coriano:2013jba ; Coriano:2018bbe ;
Anand:2019lkt ; Caloro:2022zuy ; Meltzer:2021zin , position-Mellin space
Mack:2009gy ; Mack:2009mi ; Penedones:2010ue ; Paulos:2011ie ;
Fitzpatrick:2011ia ; Kharel:2013mka ; Penedones:2019tng ; Rastelli:2016nze ;
Zhou:2017zaw ; Rastelli:2017udc ; Alday:2020dtb ; Alday:2020lbp ;
Alday:2021ajh ; Gopakumar:2016cpb ; Jepsen:2018dqp ; Jepsen:2018dqp ;
Jepsen:2019svc ; Li:2023azu ; Gadde:2022ghy ; Bianchi:2021piu ; Hoback:2020pgj
; Haldar:2019prg , and more recently momentum-Mellin space Sleight:2021iix ;
Sleight:2019hfp ; Mei:2023jkb . In this paper, our primary focus is on
studying gluon scattering within AdS. We find the advancements in flat space
scattering amplitudes, particularly those involving gluon and graviton
scattering, to be remarkably intriguing Elvang:2013cua . These developments
not only bolster experimental results at major colliders like the LHC but also
revitalize foundational quantum field theory research. A notable aspect of
gluon amplitudes in flat space is their simplicity and elegance; despite the
complexity of intermediate calculations, the final results, as epitomized by
the classic Parke-Taylor formula, are often concise and elegant Parke:1986gb .
Furthermore, these advancements unveil fascinating connections between core
physics and diverse mathematical fields Arkani-Hamed:2013jha ; Arkani-
Hamed:2017mur . Motivated by these developments, our focus is on studying
gluon scattering within AdS.

We use Mellin space as our investigative tool and it offers unique advantages.
In Mellin space, amplitudes are clearly presented as meromorphic functions of
their variables, echoing the well-understood analytic properties of the
S-Matrix in flat space. However, Mellin space has not been fully explored,
especially when examining spinning correlators Goncalves:2014rfa ;
Costa:2014kfa ; Nishida:2018opl ; Sleight:2017fpc . Our study additionally
focuses on addressing the challenging issue of higher-point correlators with
external spin, a task underscored by the limited amount of analytical work in
this area due to its technical complexity. Yet, these higher-point analysis
are important for major theoretical breakthroughs. Insights from the modern
S-matrix program show that deeper exploration of higher-point gauge and
gravity amplitude (including loop amplitudes) has greatly helped us unravel
deep mathematical structures. Hence, a thorough examination of higher-point
spinning structures in Anti-de-Sitter (AdS) space is essential to uncover
potential simplicities and mathematical insights akin to flat space scattering
amplitudes.

In addition to their relevance in Anti-de-Sitter (AdS) space, these structures
carry broader implications. Notably, they are interconnected with de Sitter
(dS) Maldacena:2002vr ; Ghosh:2014kba  aligning well with the program to
construct cosmologically relevant correlators Arkani-Hamed:2018kmz ;
Baumann:2020dch ; Sleight:2020obc ; Arkani-Hamed:2023kig . Spinning
correlators in AdS could have substantial importance in the cosmological
frontier. Moreover, specific case studies are crucial for advancing our
understanding of the still-ambiguous double copy principle in curved
spacetime. This principle is particularly important when applied to higher-
point structures, and thus, concrete examples are indispensable for its
possible formulation akin to flat space.

In this paper, we unveil a formalism anchored in embedding-space techniques to
meet our research objectives. Utilising key differential operators, we
streamline the complex calculations tied to higher-point correlators with
external spinning fields. By methodically building upon lower-point AdS
correlators, we achieve recursive computations of higher-point amplitudes in
AdS. The paper’s structure is as follows: In Section 2, we articulate the
foundational principles and techniques vital for AdS amplitude calculations.
We delve into the embedding formalism specific to AdS space and highlight the
role of Mellin space as an eigenspace for these amplitudes. We also present a
summary of our main results. Section 3 offers a comprehensive computation of
three, four, and five-point amplitudes, paving the way for subsequent, more
nuanced higher-point analysis. Here the elegant mapping between flat-space
Feynman rules and AdS begins to emerge. In Section 4, we derive a recursion
formula for $n$-point amplitudes, to assist an ambitious calculation of six-
point, seven-point, and eight-point gluon topologies. Notably, we again notice
that Mellin amplitudes for gluons strikingly parallel flat-space scattering
amplitudes, despite the complexity of intermediate calculations. This
revelation leads us to propose a remarkably streamlined map to flat space for
$n$-point gluon amplitudes. Finally, we discuss important work that can spur
from our results in Section 5.

This paper is a substantial expansion of the companion version Chu:2023pea
which we recommend to the reader who want to skip technical details and
interested in the main essence on the first reading.

##  2 Preliminaries and summary

AdS amplitude is holographically dual to Conformal Field Theory correlation
function, $\langle\mathcal{O}_{1}(P_{1})\cdots\mathcal{O}_{n}(P_{n})\rangle$
where $P_{i}$ denotes the AdS boundary coordinate where the operator
$\mathcal{O}_{i}$ is inserted. Here we provide an overview of the fundamental
ingredients and concepts involved in calculating AdS amplitudes.

###  2.1 Embedding space formalism

The calculation of Witten diagrams is markedly streamlined with the
application of the embedding formalism Penedones:2010ue .111In a seminal work
by Dirac Dirac:1936fq , it was proposed that the conformal group $SO(d+1,1)$
naturally “lives” in the embedding space $\mathbb{R}^{d+1,1}$. Here, it can be
understood as the group of linear isometries. This suggests that constraints
imposed by conformal symmetry could be as straightforward as those from
Lorentz symmetry. Also see Weinberg’s paper Weinberg:2010fx . This formalism
stands as a robust tool for the in-depth exploration and analysis of the
properties and dynamics inherent in AdS spaces. This formalism allows us to
describe an AdSd+1 space by embedding it in a higher-dimensional Minkowski
space, denoted as $\mathbb{R}^{d+1,1}$. AdS coordinate vectors $X$ satisfy the
following property,

$\displaystyle X\cdot X\equiv\eta_{MN}X^{M}X^{N}=-R^{2}.$ (2.1)

Throughout the paper we will take $R=1$. The boundary of the AdSd+1 space is
at $X\to\infty$, where (2.1) asymptotes to an equation of a light cone. It is
convenient to think of the conformal boundary of AdS as the space of null
rays.

We use $P$ to denote the fixed boundary point. Hence, $P\cdot
P\equiv\eta_{MN}P^{M}P^{N}=0$. Therefore, the distance between any two
boundary points $P_{i}$ and $P_{j}$ is defined by
$P_{ij}\equiv\left(P_{i}-P_{j}\right)^{2}=-2P_{i}\cdot P_{j}$.

###  2.2 Mellin space

Another key mathematical apparatus utilized in our study is the Mellin
space.222See Fitzpatrick:2011ia  for a nice review of Mellin Space in the
AdS/CFT. Mellin amplitudes have structural similarity to flat space momentum
space scattering amplitudes. Many researchers have demonstrated that the
Mellin representation has advantages in analyzing CFT correlation functions,
particularly within the large $N$ expansion.

The basis of Mellin space is $\prod_{i<j}P_{ij}^{-\gamma_{ij}}$, where
$\gamma_{ij}$ are called Mellin variables. The scaling dimension of this basis
for $P_{i}$ is $\sum_{j\neq i}\gamma_{ij}$. First, we focus on the scalar
cases. Expanded in Mellin space, an $n$-point amplitude can be expressed as

$\displaystyle\left\langle\prod_{i=1}^{n}\mathcal{O}_{i}(P_{i})\right\rangle=\int\left(\prod_{i<j}^{n}\frac{d\gamma_{ij}}{2\pi
i}\Gamma(\gamma_{ij})P_{ij}^{-\gamma_{ij}}\right)\prod_{i=1}^{n}\delta\Big{(}\sum_{j\neq
i}\gamma_{ij}-\Delta_{i}\Big{)}\mathscr{M}_{n}(\gamma_{ij})\ ,$ (2.2)

where $\mathscr{M}_{n}(\gamma_{ij})$ is called Mellin amplitude. Note that the
delta functions restrict the correct scaling behavior of
$\mathcal{O}_{i}(P_{i})$. For the sake of notational simplicity, we will forgo
including them in our subsequent equations.

In the context of vector fields $J^{M_{i}}(P_{i})$, our primary interest in
this paper, the amplitude takes on a slightly different form to incorporate
the indices. We can write it as

$\displaystyle\left\langle\prod_{i=1}^{n}J^{M_{i}}(P_{i})\right\rangle=\int\left(\prod_{i<j}^{n}\frac{d\gamma_{ij}}{2\pi
i}\Gamma(\gamma_{ij})P_{ij}^{-\gamma_{ij}}\right)\mathscr{M}_{n}^{M_{1}M_{2}\cdots
M_{n}}(\gamma_{ij},P_{i})\ .$ (2.3)

In this context, it is crucial to underline a subtle difference as compared to
the scalar scenario. Specifically, the Mellin amplitude
$\mathscr{M}_{n}^{M_{1}M_{2}\cdots M_{n}}(\gamma_{ij},P_{i})$ is a function
not only of the Mellin variables $\gamma_{ij}$, but also of the boundary
coordinates $P_{i}$. This is attributed to the possibility that vector Mellin
amplitude may contain $P_{i}$ with free indices.333More generally, each field
in the correlation function has a spin of $\ell_{i}$. Then, there are totally
$\sum_{i=1}^{n}\ell_{i}$ free indices in the Mellin amplitude.

###  2.3 AdS amplitudes and toolkit

Witten diagram, a powerful tool for computing amplitudes in Anti-de Sitter
space, provides a systematic approach to analyze scattering processes. It is
composed of two key elements: vertices and propagators. Vertices represent the
interaction points where particles or fields within the AdS theory come
together. They are integrated over the entire AdS space, encapsulating the
bulk interactions.

Propagators, on the other hand, come in two forms: Boundary-to-bulk
propagators connect a point on the AdS boundary to a vertex in the bulk,
capturing the information flow from the boundary into the bulk. Meanwhile,
bulk-to-bulk propagators link two vertices within the bulk, accounting for the
propagation of particles or fields between these interaction points.

#### Scalar

The boundary-to-bulk propagator for a scalar field $\mathcal{O}_{i}$ is a
function of the boundary point $P_{i}$ and the bulk point $X$, i.e.,

$\displaystyle\mathcal{E}(P_{i},X)=\frac{C_{\Delta_{i}}}{(-2P_{i}\cdot
X)^{\Delta_{i}}}\ ,\quad
C_{\Delta_{i}}=\frac{\Gamma(\Delta_{i})}{2\pi^{h}\Gamma\left(\Delta_{i}+1-h\right)}\
,$ (2.4)

where $h\equiv d/2$. To illustrate this, let’s consider the calculation of the
three-point scalar amplitude. In this case, we can compute the amplitude by
utilizing the boundary-to-bulk propagator in the following straightforward
manner:

$\displaystyle\left\langle\mathcal{O}_{1}(P_{1})\mathcal{O}_{2}(P_{2})\mathcal{O}_{3}(P_{3})\right\rangle=ig\int_{\text{AdS}}dX\
\mathcal{E}(P_{1},X)~{}\mathcal{E}(P_{2},X)\mathcal{E}(P_{3},X)\ ,$ (2.5)

where $g$ is the coupling constant. The Mellin amplitude, as it turns out (see
Appendix A for more details), is given (as shown in for instance Paulos:2011ie
) ,

$\displaystyle\mathscr{M}_{3}(P_{1},P_{2},P_{3})=ig\frac{\pi^{h}}{2}\prod_{i=1}^{3}\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+\Delta_{3}-d}{2}\right)\
.$ (2.6)

#### Vector

In this paper, we compute higher-point amplitudes, taking into account fields
with spinning degrees of freedom in both the internal propagator and external
state. The boundary-to-bulk propagator for a vector field can be obtained by
applying a differential operator to a scalar boundary-to-bulk propagator
Paulos:2011ie . These operators act as projectors, projecting the spinning
Mellin amplitude $\mathscr{M}_{n}^{M_{1}M_{2}\cdots M_{n}}$ onto a subspace
that remains conformally invariant. Specifically, for a vector field
$J^{M_{i}}(P_{i})$,

$\mathcal{E}^{M_{i}A_{i}}(P_{i},X)=\widehat{D}^{M_{i}A_{i}}\mathcal{E}(P_{i},X),$
(2.7)

where the operator $\widehat{D}^{M_{i}A_{i}}$ is defined as follows:

$\widehat{D}^{M_{i}A_{i}}=\frac{\Delta_{i}-1}{\Delta_{i}}\eta^{M_{i}A_{i}}+\frac{1}{\Delta_{i}}\frac{\partial}{\partial
P_{i}^{M_{i}}}P_{i}^{A_{i}}\ .$ (2.8)

We want to highlight that the operator $\widehat{D}^{M_{i}A_{i}}$ simplifies
the index structure of vector amplitudes, making it easier to relate to scalar
amplitudes. In anticipation of future computations and for the sake of
notational simplicity, let us introduce a concise version of the operator as
follows:

$\displaystyle\left(\prod_{i=1}^{n}\mathfrak{D}^{M_{i}A_{i}}\right)=\prod_{i=1}^{n}\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\widehat{D}^{M_{i}A_{i}}\
.$ (2.9)

#### Étude of momentum conservation analogues

We provide some properties of the differential operator given in (2.8). This
observation will be instrumental in deriving analogues of momentum
conservation, as illustrated below.

An eigenfunction of the differential operator $\widehat{D}^{M_{i}A_{i}}$ can
be expressed as $\frac{\partial}{\partial
P_{i}^{A_{i}}}F_{\delta_{i}}(P_{i})$, where $F_{\delta_{i}}(P_{i})$ denotes
any function of $P_{i}$ with the scaling dimension of $\delta_{i}$. That is,
$P_{i}\cdot\frac{\partial}{\partial
P_{i}}F_{\delta_{i}}(P_{i})=-\delta_{i}F_{\delta_{i}}(P_{i})$. Then,

$\displaystyle\widehat{D}^{M_{i}A_{i}}\frac{\partial}{\partial
P_{i}^{A_{i}}}F_{\delta_{i}}(P_{i})=\frac{\Delta_{i}-1-\delta_{i}}{\Delta_{i}}\frac{\partial}{\partial
P_{i}^{M_{i}}}F_{\delta_{i}}(P_{i})\ .$ (2.10)

Notably, when $\delta_{i}=\Delta_{i}-1$, or the scaling dimension of the
eigenfunction $\frac{\partial}{\partial P_{i}^{A_{i}}}F_{\delta_{i}}(P_{i})$
is $\Delta_{i}$, the eigenvalue is zero.

Let’s see couple of examples. Firstly, by substituting
$F_{\Delta_{i}-1}=f(\gamma_{ij})\prod_{\ell<m}\Gamma(\gamma_{\ell m})P_{\ell
m}^{-\gamma_{\ell m}}$ for some $i$ (with any function $f(\gamma_{ij})$ of
Mellin variables $\gamma_{ij}$ for all $j\neq i$) in (2.10), with
$\Delta_{i}-1=\sum_{j\neq i}\gamma_{ij}$, we deduce that

$0=\int\prod_{\ell<m}d\gamma_{\ell m}\ \widehat{D}^{M_{i}A_{i}}\sum_{k\neq
i}\big{[}P_{k,A_{i}}f(\gamma_{ij})\Gamma(\gamma_{ik}+1)P_{ik}^{-\gamma_{ik}-1}\prod_{\begin{subarray}{c}\ell<m\\\
(\ell m)\neq(ik)\end{subarray}}\Gamma(\gamma_{\ell m})P_{\ell
m}^{-\gamma_{\ell m}}\big{]}\\\ =\int\prod_{\ell<m}d\gamma_{\ell m}\
\widehat{D}^{M_{i}A_{i}}\sum_{k\neq i}\big{[}P_{k,A_{i}}f(\gamma_{ij,j\neq
k},\gamma_{ik}-1)\prod_{\ell<m}\Gamma(\gamma_{\ell m})P_{\ell
m}^{-\gamma_{\ell m}}\big{]}.$ (2.11)

In the final step we have shifted the Mellin variables,
$\gamma_{ik}\to\gamma_{ik}-1$.444So now $\sum_{j\neq
i}\gamma_{ij}=\Delta_{i}$.

As another example, by substituting
$F_{\Delta_{i_{1}}-1}=P_{i_{1},A_{i_{2}}}f(\gamma_{i_{1}j})\prod_{\ell<m}\Gamma(\gamma_{\ell
m})P_{\ell m}^{-\gamma_{\ell m}}$, for some $i_{1}$ and $i_{2}$, into (2.10),
we deduce that

$0=\int\prod_{\ell<m}d\gamma_{\ell m}\
\widehat{D}^{M_{i_{1}}A_{i_{1}}}\big{[}\eta_{A_{i_{1}}A_{i_{2}}}f(\gamma_{i_{1}j})\prod_{\ell<m}\Gamma(\gamma_{\ell
m})P_{\ell m}^{-\gamma_{\ell m}}\\\ -2P_{i_{1},A_{i_{2}}}\sum_{k\neq
i_{1}}P_{k,A_{i_{1}}}f(\gamma_{i_{1}j,j\neq
k},\gamma_{i_{1}k}-1)\prod_{\ell<m}\Gamma(\gamma_{\ell m})P_{\ell
m}^{-\gamma_{\ell m}}\big{]}\ .$ (2.12)

These identities are crucial for significantly simplifying our target
expression for higher-point functions and uncovering underlying structures.

#### Bulk-to-bulk propagators

A bulk-to-bulk propagator represents the exchange of a primary field with a
scaling dimension of $\Delta$, including its descendant fields. It is
convenient that such propagators for both scalar and spinning particles can be
written as the product of two boundary-to-bulk propagators glued together by
integration over the boundary point $Q$. This property can help us recycle the
lower-point function and obtain the higher-point function by appropriately
gluing lower-point amplitudes. For pedagogical value, we first write the
propagator associated with simpler scalar fields,

$\displaystyle\mathcal{G}_{\Delta}(X_{1},X_{2})=\int_{-i\infty}^{i\infty}\frac{dc}{2\pi
i}\frac{2c^{2}}{c^{2}-(\Delta-h)^{2}}\int_{\partial\text{AdS}}dQ\
\mathcal{E}_{h+c}(Q,X_{1})\mathcal{E}_{h-c}(Q,X_{2})\ .$ (2.13)

We can deform the integration contour in (2.13) and integrate around the pole,
e.g., $c=\Delta-h$. We subsequently get

$\displaystyle\mathcal{G}_{\Delta}(X_{1},X_{2})=\left(h-\Delta\right)\int_{\partial\text{AdS}}dQ\
\mathcal{E}_{\Delta}(Q,X_{1})\mathcal{E}_{d-\Delta}(Q,X_{2})\ .$ (2.14)

Similarly, for vector fields, the bulk-to-bulk propagator is Paulos:2011ie

$\displaystyle\mathcal{G}^{AB}_{\Delta}(X_{1},X_{2})=\int_{-i\infty}^{i\infty}\frac{dc}{2\pi
i}f_{\Delta}(c)\int_{\partial\text{AdS}}dQ\
\mathcal{E}^{MA}_{h+c}(Q,X_{1})\eta_{MN}\mathcal{E}^{NB}_{h-c}(Q,X_{2})\ ,$
(2.15)

where

$\displaystyle
f_{\Delta}(c)=\frac{4c^{2}(h^{2}-c^{2})}{\left(c^{2}-(\Delta-h)^{2}\right)^{2}}\
.$ (2.16)

In principle, the existence of second-order poles in $f_{\Delta}(c)$
complicates the calculation of the contour integral. However, we will show
that for a bulk-to-bulk propagator (one end of which is a three-vertex
connected to two external fields on the boundary), these second-order poles
simplify to first-order poles! This simplification enables easier integration
with respect to $c$ and facilitates a recursive calculation of amplitudes in
the channel with at most a single four-vertex. In summary, the structure of
vector amplitudes is similar to that of scalar fields, i.e. both can be
decomposed into products of lower-point amplitudes.

###  2.4 Summary of the main results

In this paper, we explicitly calculate the gluon Mellin amplitudes for several
diagrams, spanning from three points to eight points. In addition to detailed
calculations, this paper also serves as a repository for explicit higher-point
results. To assist the reader, here we direct the reader to the main results
of the paper.

The three-point gluon Mellin amplitude is presented in (3.2). Similarly, the
four-point amplitudes include both contact and exchange channels. The explicit
results for these are given respectively in (3.6) and (3.13), where we have
reproduced the results calculated in Paulos:2011ie .

For the five-point amplitudes, we have drastically simplified the results from
Kharel:2013mka  and expressed them in a more succinct form, as illustrated in
(3.25) and (3.35). Subsequently, we derive a recursion formula, shown in
(4.7), for $n$-point amplitudes. We then apply this formula to construct
higher-point calculations. More specifically, we present six-point amplitudes
(refer to (4.13) and (4.18)), a seven-point amplitude (refer to (4.22)), and
an eight-point amplitude (refer to (4.26)).

Besides presenting explicit novel computations, we compare our results with
their flat-space counterparts. This comparison uncovers a remarkable
resemblance between Mellin amplitudes and flat-space amplitudes, as detailed
in the dictionary presented in Table 1. In this table, the summation over the
Mellin variables is defined in (4.28). Additionally, the definitions of the
vertex factors ${\mathbf{{V}}}_{3}$ and ${\mathbf{{V}}}_{4}$ are provided in
(4.12e) and (4.25e), respectively. It is important to note that on the Mellin
side of the dictionary, an additional factor of
$\frac{\pi^{h}}{2}\prod_{i=1}^{n}\mathfrak{D}^{M_{i}A_{i}}$ should be applied.

Table 1: The correspondence between gluon flat-space amplitudes and Mellin amplitudes Description | Minkowski momentum Space | AdS Mellin Space
---|---|---
Kinematic variable | $ik_{i}$ | $2P_{i}$
Internal propagator | $\frac{i}{2\sum k_{i}\cdot k_{j}}$ | $\frac{1}{\tilde{\sum}\gamma_{ij}}$
Three-vertex coupling | $g$ | $g{\mathbf{{V}}}_{3}^{n_{a},n_{b},n_{c}}$
Four-vertex coupling | $g^{2}$ | $g^{2}{\mathbf{{V}}}_{4}^{n_{a},\cdots,n_{d}}$

##  3 Setting the stage: the three, four, and five-point gluon amplitudes

We are now poised to calculate gluon amplitudes in AdS. The non-abelian gauge
theory in Anti-de Sitter space is characterized by the action:

$\displaystyle S_{\texttt{YM}}=-\int
d^{d+1}x\sqrt{-g}\,\frac{1}{4}\,\text{Tr}\left(F_{AB}F^{AB}\right)\ ,$ (3.1)

where
$F_{AB}^{a}=\partial_{A}A_{B}^{a}-\partial_{B}A_{A}^{a}+gf^{abc}A_{A}^{b}A_{B}^{c}$
and $f^{abc}$ represent the structure constants of the gauge group. Gluon
amplitudes correspond to current correlation functions and have scaling
dimensions $\Delta_{i}=d-1$.

###  3.1 Three-point gluon amplitude

The three-point gluon Mellin amplitude shown in Figure 1, is Paulos:2011ie
555A detailed calculation can be found in Appendix A

$\mathscr{M}^{M_{1}M_{2}M_{3}}_{\texttt{3v}}=ig~{}\frac{\pi^{h}}{2}f^{a_{1}a_{2}a_{3}}\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+\Delta_{3}-d+1}{2}\right)\left(\prod_{i=1}^{3}\mathfrak{D}^{M_{i}A_{i}}\right)\mathscr{I}_{A_{1}A_{2}A_{3}}\
,$ (3.2)

where we remind the readers again that
$\prod_{i=1}^{3}\mathfrak{D}^{M_{i}A_{i}}=\prod_{i=1}^{3}\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\widehat{D}^{M_{i}A_{i}}$
and

$\begin{split}\mathscr{I}_{A_{1}A_{2}A_{3}}&=2\eta_{A_{1}A_{2}}\left(P_{1}-P_{2}\right)_{A_{3}}+\text{cyclic
permutations}\ .\end{split}$ (3.3)

Throughout this paper, we establish a beautiful correspondence between flat-
space and AdS amplitudes. As evident, already from the simple three-point
function, the Mellin amplitude remarkably resembles its flat-space
counterpart, obtained from the Feynman rules,

$\begin{split}\mathscr{A}_{\texttt{3v},A_{1}A_{2}A_{3}}&=-gf^{a_{1}a_{2}a_{3}}\eta_{A_{1}A_{2}}(k_{1}-k_{2})_{A_{3}}+\text{cyclic
permutations}\ .\end{split}$ (3.4)

This similarity becomes apparent when the momenta $ik_{i,A}$ are mapped to
$2P_{i,A}$ and the three-vertex coupling constant $g$ is associated with $g\
{\mathbf{{V}}}_{3}^{0,0,0}$, where

$\displaystyle{\mathbf{{V}}}_{3}^{0,0,0}\equiv\Gamma(d-1)\ .$ (3.5)

The three null arguments in (3.5) indicate that the three-vertex is linked to
three boundary-to-bulk propagators. This correspondence holds, taking into
account the differential operators $\prod_{i=1}^{3}\mathfrak{D}^{M_{i}A_{i}}$
and a constant factor $\frac{\pi^{h}}{2}$.

Figure 1:  From left to right, (a) the three gluon amplitude, (b) the contact
diagram of the four gluon amplitude, and (c) the $s$-channel representation of
the four gluon amplitude.

###  3.2 Four-point gluon amplitudes

####  3.2.1 Contact diagram

For the four-point contact diagram (Figure 1), the Mellin amplitude is
Kharel:2013mka

$\mathscr{M}^{M_{1}M_{2}M_{3}M_{4}}_{\texttt{Contact}}=-ig^{2}\frac{\pi^{h}}{2}\left(f^{a_{1}a_{4}b^{\prime}}f^{a_{2}a_{3}b^{\prime}}+f^{a_{1}a_{3}b^{\prime}}f^{a_{2}a_{4}b^{\prime}}\right)\left(\prod_{i=1}^{4}\mathfrak{D}^{M_{i}A_{i}}\right)\\\
\times\Gamma\left(\frac{\sum_{i=1}^{4}\Delta_{i}-d}{2}\right)\eta_{A_{1}A_{2}}\eta_{A_{3}A_{4}}+\text{cyclic
perm. of (123)}\ .$ (3.6)

Note that it is also the same as the flat-space counterpart from the Feynman
rules in Yang-Mills theory,

$\displaystyle\mathscr{A}_{\texttt{Contact}}=-ig^{2}\left(f^{a_{1}a_{4}b^{\prime}}f^{a_{2}a_{3}b^{\prime}}+f^{a_{1}a_{3}b^{\prime}}f^{a_{2}a_{4}b^{\prime}}\right)\eta_{A_{1}A_{2}}\eta_{A_{3}A_{4}}+\text{cyclic
perm. of (123)}\ ,$ (3.7)

up to $\frac{\pi^{h}}{2}\prod_{i=1}^{4}\mathfrak{D}^{M_{i}A_{i}}$, with the
identification of the four-vertex coupling constant, i.e.,
$g^{2}{\mathbf{{V}}}_{4}^{0,0,0,0}$, where

$\displaystyle{\mathbf{{V}}}_{4}^{0,0,0,0}\equiv\Gamma\left(\frac{3d-4}{2}\right)\
.$ (3.8)

####  3.2.2 Exchange diagram

The $s$-channel is shown in Figure 1. The amplitude can be expressed in terms
of the three-point function by utilizing the factorization in (2.15),

$\Big{\langle}\prod_{i=1}^{4}J^{M_{i}}(P_{i})\Big{\rangle}_{\texttt{Exch}}=\int_{-i\infty}^{i\infty}\frac{dc}{2\pi
i}f_{\Delta}(c)\int_{\partial\text{AdS}}dQ\
\big{\langle}J^{M_{1}}(P_{1})J^{M_{2}}(P_{2})J_{h+c}^{M}(Q)\big{\rangle}\eta_{MN}\\\
\times\big{\langle}J_{h-c}^{N}(Q)J^{M_{3}}(P_{3})J^{M_{4}}(P_{4})\big{\rangle}\
,$ (3.9)

where the subscripts $h\pm c$ of the exchange vector field indicate its
scaling dimension. The integration over $Q$ can be performed by employing
Symanzik’s formula Symanzik:1972wj ,

$\displaystyle\int_{\partial\text{AdS}}dQ\prod_{i=1}^{n}\Gamma(\ell_{i})\left(-2P_{i}\cdot
Q\right)^{-\ell_{i}}=\pi^{h}\int\left(\prod_{i<j}^{n}\frac{d\gamma_{ij}}{2\pi
i}\Gamma(\gamma_{ij})P_{ij}^{-\gamma_{ij}}\right)\prod_{i=1}^{n}\delta\left(\sum_{j\neq
i}^{n}\gamma_{ij}-\ell_{i}\right)$ (3.10)

(note, $\sum_{i=1}^{n}\ell_{i}=d$). For review, we refer the reader to
Appendix B.

For $Q$’s with free indices in (3.9), we replace all the occurrences by
$P_{i}$’s employing (2.10) Paulos:2011ie ; Kharel:2013mka . Therefore, we get

$\langle J^{M_{1}}(P_{1})J^{M_{2}}(P_{2})J^{M}_{h\pm
c}(Q)\rangle=ig\frac{\pi^{h}}{2}f^{a_{1}a_{2}b}\left(\prod_{i=1}^{2}\mathfrak{D}^{M_{i}A_{i}}\right)\frac{C_{h\pm
c}}{\Gamma(h\pm c)}\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+h\pm
c-d+1}{2}\right)\\\ \times\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-(h\pm
c)+1}{2}\right)\Gamma\left(\frac{\Delta_{1}-\Delta_{2}+(h\pm
c)+1}{2}\right)\Gamma\left(\frac{-\Delta_{1}+\Delta_{2}+h\pm
c-1}{2}\right)\frac{h\pm c-1}{h\pm c}\\\
\times\Big{\\{}X_{12}^{M}\Big{\\}}\times
P_{12}^{-\frac{\Delta_{1}+\Delta_{2}-(h\pm c)+1}{2}}(-2P_{1}\cdot
Q)^{-\frac{\Delta_{1}-\Delta_{2}+h\pm c+1}{2}}(-2P_{2}\cdot
Q)^{-\frac{-\Delta_{1}+\Delta_{2}+h\pm c-1}{2}}+(1\leftrightarrow 2)\ ,$
(3.11)

where

$\Big{\\{}X_{ij}^{M}\Big{\\}}\equiv
2\left(\eta_{A_{i}A_{j}}P_{i}^{M}-2\delta^{M}_{A_{i}}P_{i,A_{j}}\right)-(i\leftrightarrow
j)\ .$ (3.12)

Note that in (3.11) we have explicitly performed the action of the
differential operator $\widehat{D}^{MA}_{h\pm c}$.

Let’s take a moment to scrutinize the factor $\frac{h\pm c-1}{h\pm c}$ in
(3.11). One can see that it possesses simple zeros at $c=\pm(h+1)$.
Importantly, with $\Delta=d-1$, the simple zeros are at the same position as
the double poles of $f_{\Delta}(c)$. Therefore, the poles reduce to simple
poles.

Integrating around one of the simple poles, say $c=h+1$ without loss of
generality, one get the Mellin amplitude for the $s$-channel Paulos:2011ie

$\mathscr{M}^{M_{1}M_{2}M_{3}M_{4}}_{\texttt{Exch}}=-g^{2}~{}\frac{\pi^{h}}{2}f^{a_{1}a_{2}b}f^{a_{3}a_{4}b}\left(\prod_{i=1}^{4}\mathfrak{D}^{M_{i}A_{i}}\right)\\\
\times\left(\sum_{n=0}^{\infty}\frac{\Big{\\{}X_{12}^{M}\Big{\\}}{\mathbf{{V}}}_{3}^{n,0,0}\times{\mathbf{{V}}}_{3}^{n,0,0}\Big{\\{}X_{34,M}\Big{\\}}}{4n!~{}\Gamma\left(\frac{d}{2}+n\right)\left(\gamma_{12}-\frac{d}{2}+n\right)}\right),$
(3.13)

where $(a)_{n}\equiv a(a+1)(a+2)\cdots(a+n-1)$ is the Pochhammer symbol, the
contribution of the boundary points is $\\{X_{ij}^{M}\\}$ and

$\displaystyle{\mathbf{{V}}}_{3}^{n,0,0}\equiv\left(\frac{d}{2}-n\right)_{n}\Gamma\left(d-1\right)$
(3.14)

stands for the contribution of the three-vertices connecting to one bulk-to-
bulk propagator. Note that with $n=0$, ${\mathbf{{V}}}_{3}^{n,0,0}$ reduces to
the expression defined in (3.5).

The series of simple poles, $\gamma_{12}=\frac{d}{2}-n$, comes from the Gamma
function $\Gamma(\gamma_{12})$ in (3.10), where $\gamma_{12}$ should be
shifted in order to incorporate the power of $P_{12}$ in (3.11).
Interestingly, for even $d$, the infinite sum is cut-off at $n=\frac{d}{2}$.
So,

$\displaystyle\sum_{n=0}^{\infty}\frac{{\mathbf{{V}}}_{3}^{n,0,0}\times{\mathbf{{V}}}_{3}^{n,0,0}}{4n!\Gamma\left(\frac{d}{2}+n\right)\left(\gamma_{12}-\frac{d}{2}+n\right)}=$
$\displaystyle\frac{\big{(}\Gamma\left(d-1\right)\big{)}^{2}}{4\Gamma\left(\frac{d}{2}\right)\left(\gamma_{12}-\frac{d}{2}\right)}+\frac{\left(\Gamma\left(d-1\right)\left(\frac{d}{2}-1\right)\right)^{2}}{4\Gamma\left(\frac{d}{2}+1\right)\left(\gamma_{12}-\frac{d}{2}+1\right)}$
(3.15)
$\displaystyle+\cdots\cdots\cdots+\frac{\Gamma\left(d-1\right)\left(\left(\frac{d}{2}-1\right)!\right)^{2}}{4(\gamma_{12}-1)}\
.$

Similar to the three-point and contact examples, one can map the Mellin
amplitude (3.13) from the flat-space amplitude, which from the Feynman rules
is

$\mathscr{A}_{\texttt{Exch}}=g^{2}f^{a_{1}a_{2}b^{\prime}}f^{a_{3}a_{4}b^{\prime}}\frac{i}{(k_{1}+k_{2})^{2}}\left(\eta_{A_{1}A_{2}}k_{1,N}-2\eta_{A_{1}N}k_{1,A_{2}}-(1\leftrightarrow
2)\right)\\\
\times\left(\eta_{A_{3}A_{4}}k_{3}^{N}-2\eta_{A_{3}}^{N}k_{3,A_{4}}-(3\leftrightarrow
4)\right)\ ,$ (3.16)

by replacing the momenta $ik_{i}\to 2P_{i}$, the propagator

$\displaystyle\frac{i}{(k_{1}+k_{2})^{2}}\to\frac{1}{4n!\Gamma\left(\frac{d}{2}+n\right)\left(\gamma_{12}-\frac{d}{2}+n\right)}$
(3.17)

(with integer $n$ to be summed from 0 to the infinity) and the three-vertex
coupling constant $g\to g\ {\mathbf{{V}}}_{3}^{n,0,0}$.

It is worth mentioning that there does not exist a unique way to express
Mellin amplitudes, since it is a part of the integrand in the full correlation
function (2.3). For example, the term $P_{1}\cdot P_{3}$ can be absorbed into
the Mellin basis with a shift of Mellin variable
$\gamma_{13}\to\gamma_{13}+1$, as in Paulos:2011ie . However, we express the
Mellin amplitude in a transparent way to show resemblance to the flat-space
amplitude.

In this work, we aim to investigate Mellin gluon amplitude beyond four-point
functions for different topologies. As an initial demonstration, we focus our
attention on the five-point amplitudes.

###  3.3 Five-point gluon amplitudes

In this section, we explore five-point gluon AdS amplitudes in Yang-Mills
theory, denoted as $\left\langle J^{M_{1}}(P_{1})\cdots
J^{M_{5}}(P_{5})\right\rangle$. These amplitudes encompass channels with two
distinct types of topologies. The calculation for each five-point function
involves three major steps: 1) Factorizing the five-point correlation function
into a three-point function and a four-point function, implementing the
integration over $Q$ using Symanzik’s formula. 2) Simplifying the double poles
in the integration over $c$. 3) Performing the integration over the Mellin
variables in the four-point function. These steps enable the computation of
explicit expressions and facilitate the mapping between Mellin amplitudes and
flat-space amplitudes.

####  3.3.1 Channel with a three-vertex and a four-vertex

Figure 2:  Five-point channel consisting of contact and three-point vertex
interactions

We initiate our investigation by focusing on a topology, wherein a bulk-to-
bulk propagator establishes a connection between a four-vertex with points
$P_{1}$, $P_{2}$, and $P_{3}$, and a three-vertex involving points $P_{4}$ and
$P_{5}$. This configuration is visually represented in Figure 2. Using the
factorization property given by equation (2.15), the amplitude can be written
as:
$\left\langle\prod_{i=1}^{5}J^{M_{i}}(P_{i})\right\rangle_{\texttt{3v4v}}=\int_{-i\infty}^{i\infty}\frac{dc}{2\pi
i}f_{\Delta}(c)\int_{\partial\text{AdS}}dQ\left\langle\prod_{i=1}^{3}{J}^{M_{i}}(P_{i})J^{M}_{h+c}(Q)\right\rangle_{\texttt{Contact}}\\\
\times\eta_{MN}\left\langle
J^{N}_{h-c}(Q)J^{M_{4}}(P_{4})J^{M_{5}}(P_{5})\right\rangle\ .$ (3.18a)

Let us first look at the expression for contact diagram (3.6). We will
identify $P_{4}$ in (3.6) with $Q$ to match the factorized contact diagram as
shown in Figure 2. By acting with the differential operator
$\widehat{D}^{M_{4}A_{4}}$, we substitute $P_{4}$ with a free index into
(3.6). Utilizing Equation (2.10), we can succinctly recasts the four-point
contact diagram as follows Kharel:2013mka :

$\left\langle\prod_{i=1}^{4}J^{M_{i}}(P_{i})\right\rangle_{\texttt{Contact}}=-ig^{2}~{}\frac{\pi^{h}}{2}\left(f^{a_{1}a_{4}b^{\prime}}f^{a_{2}a_{3}b^{\prime}}+f^{a_{1}a_{3}b^{\prime}}f^{a_{2}a_{4}b^{\prime}}\right)\left(\prod_{i=1}^{3}\mathfrak{D}^{M_{i}A_{i}}\right)\frac{C_{4}}{\Gamma(\Delta_{4})}\\\
\times\int\prod_{k<\ell}^{4}\frac{d\gamma_{k\ell}^{\prime}}{2\pi
i}\Gamma(\gamma_{k\ell}^{\prime})P_{k\ell}^{-\gamma_{k\ell}^{\prime}}\left(\frac{\Delta_{4}-1}{\Delta_{4}}\eta_{A_{3}}^{M_{4}}-\frac{1}{\Delta_{4}}\left(P_{1,A_{3}}\left(P_{1}^{M_{4}}+P_{2}^{M_{4}}+P_{3}^{M_{4}}\right)+(1\leftrightarrow
2)\right)\right)\\\
\times\Gamma\left(\frac{\sum_{i=1}^{4}\Delta_{i}-d}{2}\right)\eta_{A_{1}A_{2}}+\text{cyclic
perm. of (123)}\ .$ (3.18b)

Starting from the constraint imposed by the Mellin variables, denoted as
$\Delta_{i}^{\prime}\equiv\sum_{j\neq i}\gamma_{ij}^{\prime}$ (_note that
$\Delta_{i}^{\prime}$ is not necessarily $\Delta_{i}$. For example, the term
$P_{1A_{3}}P_{1}^{M_{4}}$ has scaling dimension of $-2$ for $P_{1}$. To
compensate that, $\Delta_{1}^{\prime}=\Delta_{1}+2$_), we have the opportunity
to dispense with $\gamma^{\prime}_{i4}$ to arrive at the following constraint:

$2\sum_{i<j}^{3}\gamma_{ij}^{\prime}=\Delta_{1}^{\prime}+\Delta_{2}^{\prime}+\Delta_{3}^{\prime}-\Delta_{4}^{\prime}\
.$ (3.18c)

We now transition to substituting $P_{4}$ with the integration variable $Q$ in
the existing formulation.

Next, we proceed by substituting (3.11) and (3.18b) into (3.18a). Employing
the Symanzik’s formula (3.10), we obtain the Mellin amplitude,

$\mathscr{M}^{M_{1}M_{2}M_{3}M_{4}M_{5}}_{\texttt{3v4v}}=g^{3}\frac{\pi^{h}}{2}\left(f^{a_{1}bb^{\prime}}f^{a_{2}a_{3}b^{\prime}}+f^{a_{1}a_{3}b^{\prime}}f^{a_{2}bb^{\prime}}\right)f^{ba_{4}a_{5}}\int_{-i\infty}^{i\infty}\frac{dc}{2\pi
i}\frac{c^{2}-(h-1)^{2}}{\left(c^{2}-(\Delta-h)^{2}\right)^{2}}\\\
\times\left(\prod_{i=1}^{5}\mathfrak{D}^{M_{i}A_{i}}\right)\frac{1}{2\Gamma(c)\Gamma(-c)}\Gamma\left(\frac{\Delta_{4}+\Delta_{5}+h-c-d+1}{2}\right)\Gamma\left(\frac{\Delta_{4}+\Delta_{5}-h+c+1}{2}\right)\\\
\times\
\frac{\Gamma\left(\gamma_{45}-\frac{\Delta_{4}+\Delta_{5}-h+c+1}{2}\right)}{\Gamma(\gamma_{45})}\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+\Delta_{3}+h+c-d}{2}\right)\\\
\times\int\prod_{k<\ell}^{3}\frac{d\gamma_{k\ell}^{\prime}}{2\pi
i}\delta\left(\sum_{k<\ell}^{3}\gamma_{k\ell}^{\prime}+\frac{h+c-\Delta^{\prime}_{1}-\Delta^{\prime}_{2}-\Delta^{\prime}_{3}}{2}\right)\prod_{i<j}^{3}\frac{\Gamma\left(\gamma_{ij}-\gamma_{ij}^{\prime}\right)\Gamma(\gamma_{ij}^{\prime})}{\Gamma(\gamma_{ij})}\eta_{A_{1}A_{2}}\\\
\times\Big{\\{}X_{45,M}\Big{\\}}\times\left\\{\eta_{A_{3}}^{M}-\frac{2}{h+c-1}\left(P_{1,A_{3}}\left(P_{1}^{M}+P_{2}^{M}+P_{3}^{M}\right)+(1\leftrightarrow
2)\right)\right\\}+\text{cyclic perm. of (123)}\ .$ (3.19)

Note again that $\Delta_{i}^{\prime}$ depends on the scaling dimension of each
term in the last line. In particular, for a term which has scaling dimension
of $\delta_{i}$ in $P_{i}$, $\Delta_{i}^{\prime}=\Delta_{i}-\delta_{i}$.

Indeed, the expression (3.19) is very complicated and our goal is to simplify
it further. First, we bring the external fields on shell, i.e.
$\Delta_{i}=d-1$. The product of the $P_{4,M}$ term in $\\{X_{45,M}\\}$ and
the second term in the second curly bracket of (3.19) gives

$\displaystyle\eta_{A_{4}A_{5}}P_{4,M}\left(P_{1,A_{3}}\left(P_{1}^{M}+P_{2}^{M}+P_{3}^{M}\right)+(1\leftrightarrow
2)\right)\to\eta_{A_{4}A_{5}}P_{1,A_{3}}(d-1-\gamma_{45})+(1\leftrightarrow
2)\ ,$ (3.20)

where we have shifted the Mellin variables (e.g.
$\Gamma(\gamma_{14})P_{14}^{-\gamma_{14}+1}\to\gamma_{14}\Gamma(\gamma_{14})P_{14}^{-\gamma_{14}}$)
and used that $\sum_{i=1}^{3}\gamma_{i4}+\gamma_{45}=\Delta_{4}=d-1$. Since
(3.20) is symmetric under the interchange of labels $4$ and $5$, its
contribution vanishes due to the antisymmetric property of $\\{X_{45,M}\\}$.

The contribution from the multiplication of the $P_{4,A_{5}}$ term in
$\\{X_{45,M}\\}$ and the second term inside the second curly bracket of (3.19)
also vanishes due to the antisymmetry under $4\leftrightarrow 5$. To see this,
we can use (2.12) with

$\displaystyle
f(\gamma_{45})=\frac{\Gamma\left(\gamma_{45}-\frac{\Delta_{4}+\Delta_{5}-h+c+1}{2}\right)}{\Gamma(\gamma_{45})}\
,$ (3.21a) and get $\displaystyle
f(\gamma_{45})\left(\sum_{i=1}^{3}P_{i,A_{4}}\right)P_{4,A_{5}}=f(\gamma_{45})\frac{1}{2}\eta_{A_{4}A_{5}}-f(\gamma_{45}-1)P_{4,A_{5}}P_{5,A_{4}}+\cdots\
,$ (3.21b)

where $\cdots$ denotes the term vanishing upon the action of
$\widehat{D}^{M_{4}A_{4}}$. Now it is clear that (3.21b) exhibits symmetry
under $4\leftrightarrow 5$. Therefore, the antisymmetric property of
$\\{X_{45,M}\\}$ results in a total cancellation. As a result of the above
observations, we find that the second term inside the second curly bracket of
(3.19) can be neglected.

We also note that with $\Delta=d-1$, the factor

$\displaystyle\frac{c^{2}-(h-1)^{2}}{(\left(c^{2}-(\Delta-h)^{2}\right)^{2}}=\frac{1}{c^{2}-(h-1)^{2}}\
,$ (3.22)

which gives simple poles at $c=\pm(h-1)$. Therefore, with the integration of
$c$ around the pole $h-1$, (3.19) can be simplified to:

$\mathscr{M}^{M_{1}M_{2}M_{3}M_{4}M_{5}}_{\texttt{3v4v}}=g^{3}\frac{\pi^{h}}{2}\left(f^{a_{1}bb^{\prime}}f^{a_{2}a_{3}b^{\prime}}+f^{a_{1}a_{3}b^{\prime}}f^{a_{2}bb^{\prime}}\right)f^{ba_{4}a_{5}}\left(\prod_{i=1}^{5}\mathfrak{D}^{M_{i}A_{i}}\right)\frac{\Gamma\left(d-1\right)}{4}\\\
\times\frac{\Gamma\left(\gamma_{45}-d+1\right)}{\Gamma(\gamma_{45})\Gamma\left(1-\frac{d}{2}\right)}\int\left(\prod_{k<\ell}^{3}\frac{d\gamma_{k\ell}^{\prime}}{2\pi
i}\frac{\Gamma\left(\gamma_{k\ell}-\gamma_{k\ell}^{\prime}\right)\Gamma(\gamma_{k\ell}^{\prime})}{\Gamma(\gamma_{k\ell})}\right)\delta\left(\sum_{k<\ell}^{3}\gamma_{k\ell}^{\prime}-d+1\right)\\\
\times\Gamma\left(\frac{3d-4}{2}\right)\eta_{A_{1}A_{2}}\Big{\\{}X_{45,A_{3}}\Big{\\}}+\text{cyclic
perm. of (123)}\ .$ (3.23)

After integrating (3.23) over the Mellin variables $\gamma_{k\ell}^{\prime}$,
for $1\leq k,\ell\leq 3$, we encounter poles at
$\gamma_{k\ell}^{\prime}=\gamma_{k\ell}+n_{k\ell}$, with any non-negative
integers $n_{k\ell}$. However, because of the presence of a delta function,
one of these pole terms does not get integrated out.666Note that the scalar
case has a similar construction Fitzpatrick:2011ia .

To compare the expression (3.23) to flat-space amplitude, we need to simplify
it even further. We begin by considering the integral term in the second line
of (3.23): $\int\left(\prod_{k<\ell}^{3}\frac{d\gamma_{k\ell}^{\prime}}{2\pi
i}\frac{\Gamma\left(\gamma_{k\ell}-\gamma_{k\ell}^{\prime}\right)\Gamma(\gamma_{k\ell}^{\prime})}{\Gamma(\gamma_{k\ell})}\right)\delta\left(\sum_{k<\ell}^{3}\gamma_{k\ell}^{\prime}-d+1\right)$
(3.24a) Through the integration around the poles
$\gamma_{k\ell}^{\prime}=\gamma_{k\ell}+n_{k\ell}$, this term reduces to:
$\sum_{m=0}^{\infty}\frac{1}{\sum_{i<j}^{3}\gamma_{ij}-d+1+m}\sum_{\sum_{i<j}^{3}n_{ij}=m}(-1)^{m}\frac{(\gamma_{12})_{n_{12}}(\gamma_{13})_{n_{13}}(\gamma_{23})_{n_{23}}}{n_{12}!n_{13}!n_{23}!}$
(3.24b) Further simplification leads it to:
$\sum_{m=0}^{\infty}\frac{(-1)^{m}}{\sum_{i<j}^{3}\gamma_{ij}-d+1+m}\frac{\left(\sum_{i<j}^{3}\gamma_{ij}\right)_{m}}{m!}=\sum_{m=0}^{\infty}\frac{(-1)^{m}\left(d-1-m\right)_{m}}{m!(\sum_{i<j}^{3}\gamma_{ij}-d+1+m)}$
(3.24c) At the pole of (3.24c), $\sum_{i<j}^{3}\gamma_{ij}=d-1-m$, from the
constraint $\sum_{j\neq i}\delta_{ij}=\Delta_{i}^{\prime}$ on the Mellin
variables and $\Delta_{i}=d-1$, we have
$\gamma_{45}=\frac{\Delta_{4}+\Delta_{5}+1-\sum_{i=1}^{3}\Delta_{i}+2\sum_{i<j}^{3}\gamma_{ij}}{2}=\frac{d}{2}-m\
.$ (3.24d) Hence, the first term in the second line of (3.23) becomes
$\begin{split}\frac{\Gamma\left(\gamma_{45}-d+1\right)}{\Gamma\left(\gamma_{45}\right)\Gamma\left(1-\frac{d}{2}\right)}=\frac{(-1)^{m}\left(\frac{d}{2}-m\right)_{m}}{\Gamma\left(\frac{d}{2}+m\right)}\
.\end{split}$ (3.24e)

Finally, the Mellin amplitude becomes remarkably simple:

$\mathscr{M}^{M_{1}\cdots
M_{5}}_{\texttt{3v4v}}=g^{3}\frac{\pi^{h}}{2}\left(f^{a_{1}bb^{\prime}}f^{a_{2}a_{3}b^{\prime}}+f^{a_{1}a_{3}b^{\prime}}f^{a_{2}bb^{\prime}}\right)f^{ba_{4}a_{5}}\left(\prod_{i=1}^{5}\mathfrak{D}^{M_{i}A_{i}}\right)\\\
\times\sum_{m=0}^{\infty}\frac{\Big{\\{}X_{45,A_{3}}\Big{\\}}{\mathbf{{V}}}_{3}^{m,0,0}\times{\mathbf{{V}}}_{4}^{m,0,0,0}}{4m!\Gamma\left(\frac{d}{2}+m\right)(\gamma_{45}-\frac{d}{2}+m)}\eta_{A_{1}A_{2}}+\text{cyclic
perm. of (123)}\ ,$ (3.25)

where we have defined

$\displaystyle{\mathbf{{V}}}_{4}^{m,0,0,0}\equiv(d-1-m)_{m}\Gamma\left(\frac{3d-4}{2}\right)\
.$ (3.26)

Note that with $m=0$, it reduces to ${\mathbf{{V}}}_{4}^{0,0,0,0}$ given in
(3.8). We should remark that the summation over $m$ is truncated at $m=d-1$
for odd values of $d$, and at $m=\min\\{d-1,\frac{d}{2}\\}$ for even $d$. With
this stipulation in place, we are now in an ideal position to compare (3.25)
with its corresponding flat-space expression. From the Feynman rules,

$\mathscr{A}_{\texttt{3v4v}}=ig^{3}\left(f^{a_{1}bb^{\prime}}f^{a_{2}a_{3}b^{\prime}}+f^{a_{1}a_{3}b^{\prime}}f^{a_{2}bb^{\prime}}\right)f^{ba_{4}a_{5}}\left(\eta_{A_{4}A_{5}}k_{4,A_{3}}-2\eta_{A_{3}A_{4}}k_{4A_{5}}-(4\leftrightarrow
5)\right)\\\ \times\eta_{A_{1}A_{2}}\frac{i}{(k_{4}+k_{5})^{2}}+\text{cyclic
perm. of (123)}\ .$ (3.27)

One can immediately see that, the simplification seen at three and four-point
gluon amplitudes, as elaborated in Section 3 carries for the higher-point
structure with three-vertex and four-vertex topology. Specifically, we find
that the Mellin amplitude in the current channel can be straightforwardly
derived from its flat-space counterpart through a well-defined set of
substitutions. Explicitly, for any momentum term $ik_{i}$, it should be
replaced by $2P_{i}$. For the propagator,

$\frac{i}{(k_{4}+k_{5})^{2}}\to\frac{1}{4\left(\gamma_{45}-\frac{d}{2}+m\right)\Gamma\left(\frac{d}{2}+m\right)m!}\
,$ (3.28)

with $m$ to be summed over. For the three- and four-vertex coupling constant,

$\displaystyle g\mapsto g\ {\mathbf{{V}}}_{3}^{m,0,0}\ \quad\text{and}\quad
g^{2}\mapsto g^{2}\ {\mathbf{{V}}}_{4}^{m,0,0,0}\ ,$ (3.29)

respectively. In summary, our exhaustive computational analysis reveals a
striking simplification in the Mellin amplitude associated with the five-point
function. In the ensuing section, we will delve into the other topological
configurations that constitute this five-point function.

####  3.3.2 Channel with three three-vertices

Figure 3:  The channel of five-point gluon amplitude for (3.30a)

In this part of the paper, we focus on the other five-point channel
configuration where there are three-vertices, one links $P_{1}$, $P_{2}$ with
a bulk-to-bulk propagator, one links $P_{4}$, $P_{5}$ with another bulk-to-
bulk propagator, and the other links $P_{3}$ with the two bulk-to-bulk
propagators. This diagram depicted in Figure 3. By employing the relationship
specified in (2.15), we can derive the corresponding amplitude for this
configuration,

$\left\langle\prod_{i=1}^{5}J^{M_{i}}(P_{i})\right\rangle_{\texttt{3v3v3v}}=\int_{-i\infty}^{i\infty}\frac{dc}{2\pi
i}f_{\Delta}(c)\int_{\partial\text{AdS}}dQ\left\langle\prod_{i=1}^{3}J^{M_{i}}(P_{i})J^{M}_{h+c}(Q)\right\rangle_{\texttt{Exch}}\\\
\times\eta_{MN}\left\langle
J^{N}_{h-c}(Q)J^{M_{4}}(P_{4})J^{M_{5}}(P_{5})\right\rangle\ .$ (3.30a) The
exchange contribution, with $\Delta_{i}=d-1$, is given by (3.13). Performing
the action of $\widehat{D}^{M_{4}A_{4}}$ explicitly, one rewrite the
expression as
$\bigg{\langle}\prod_{i=1}^{4}J^{M_{i}}(P_{i})\bigg{\rangle}_{{\texttt{Exch}}}=-g^{2}f^{a_{1}a_{2}b^{\prime}}f^{a_{3}a_{4}b^{\prime}}\frac{\pi^{h}}{2}\left(\prod_{i=1}^{3}\mathfrak{D}^{M_{i}A_{i}}\right)\frac{C_{4}}{\Gamma(\Delta_{4})}\int\prod_{k<\ell}^{4}\frac{d\gamma_{k\ell}^{\prime}}{2\pi
i}\Gamma(\gamma_{k\ell}^{\prime})P_{k\ell}^{-\gamma_{k\ell}^{\prime}}\\\
\times\left(\sum_{n=0}^{\infty}\frac{\left(\Gamma\left(d-1\right)(\frac{d}{2}-n)_{n}\right)^{2}}{4n!\
\Gamma\left(\frac{d}{2}+n\right)\left(\gamma^{\prime}_{12}-\frac{d}{2}+n\right)}\right)\left(\frac{\Delta_{4}-1}{\Delta_{4}}\eta^{M_{4}A_{b}}\mathscr{G}_{A_{1}A_{2}A_{3}A_{b}}+\frac{2}{\Delta_{4}}\bigg{(}P_{1}^{M_{4}}+P_{2}^{M_{4}}+P_{3}^{M_{4}}\bigg{)}\mathscr{H}_{A_{1}A_{2}A_{3}}\right)\
,$ (3.30b) where
$\mathscr{G}_{A_{1}A_{2}A_{3}A_{b}}=\Big{\\{}X_{12}^{A_{b^{\prime}}}\Big{\\}}\Big{\\{}X_{bb^{\prime},A_{3}}\Big{\\}}\
,\ P_{b}\equiv-P_{1}-P_{2}-P_{3}\ ,\ P_{b^{\prime}}\equiv P_{1}+P_{2}\ ,$
(3.30c) and $\mathscr{H}_{A_{1}A_{2}A_{3}}$ is obtained from the product
$P_{4}^{A_{b}}\mathscr{G}_{A_{1}A_{2}A_{3}A_{b}}$ followed by the elimination
of the $P_{4}$ dependence using (2.11) and (2.12).

After substituting (3.11) and (3.30b) into (3.30a), we proceed to integrate
over $Q$ by using the Symanzik’s formula (3.10). Upon completing the
integration, the resulting expression yields the Mellin amplitude,

$\mathscr{M}^{M_{1}\cdots
M_{5}}_{\texttt{3v3v3v}}=-ig^{3}f^{a_{1}a_{2}b^{\prime}}f^{a_{3}bb^{\prime}}f^{ba_{4}a_{5}}\frac{\pi^{h}}{2}\int_{-i\infty}^{i\infty}\frac{dc}{2\pi
i}\frac{c^{2}-(h-1)^{2}}{\left(c^{2}-(\Delta-h)^{2}\right)^{2}}\left(\prod_{i=1}^{5}\mathfrak{D}^{M_{i}A_{i}}\right)\frac{1}{2\Gamma(c)\Gamma(-c)}\\\
\times\Gamma\left(\frac{\Delta_{4}+\Delta_{5}+h-c-d+1}{2}\right)\Gamma\left(\frac{\Delta_{4}+\Delta_{5}-h+c+1}{2}\right)\frac{\Gamma\left(\gamma_{45}-\frac{\Delta_{4}+\Delta_{5}-h+c+1}{2}\right)}{\Gamma(\gamma_{45})}\\\
\times\int\left(\prod_{k<\ell}^{3}\frac{d\gamma_{k\ell}^{\prime}}{2\pi
i}\frac{\Gamma\left(\gamma_{k\ell}-\gamma_{k\ell}^{\prime}\right)\Gamma(\gamma_{k\ell}^{\prime})}{\Gamma(\gamma_{k\ell})}\right)\delta\left(\sum_{k<\ell}^{3}\gamma_{k\ell}^{\prime}+\frac{h+c-\sum_{i=1}^{3}\Delta_{i}^{\prime}}{2}\right)\\\
\times\sum_{n=0}^{\infty}\frac{\left(\Gamma\left(d-1\right)(-\frac{d}{2}+1)_{n}\right)^{2}}{4n!\Gamma\left(\frac{d}{2}+n\right)\left(\gamma_{12}^{\prime}-\frac{d}{2}+n\right)}\Big{\\{}X_{45}^{M}\Big{\\}}\Big{\\{}\mathscr{G}_{A_{1}A_{2}A_{3}M}+\frac{2}{h+c-1}(P_{1}+P_{2}+P_{3})_{M}\mathscr{H}_{A_{1}A_{2}A_{3}}\Big{\\}}\
.$ (3.31)

Note again that here $\Delta_{i}^{\prime}$ is not necessarily $\Delta_{i}$.
From the expression of $\mathscr{G}_{A_{1}A_{2}A_{3}M}$, (3.30c),
$\sum_{i=1}^{3}\Delta_{i}^{\prime}=\sum_{i=1}^{3}\Delta_{i}+2$ for the
$\mathscr{G}$ term, and
$\sum_{i=1}^{3}\Delta_{i}^{\prime}=\sum_{i=1}^{3}\Delta_{i}+4$ for the
$\mathscr{H}$ term.

Now we simplify the expression (3.31) further. Taking $\Delta_{i}=d-1$, the
result obtained by multiplying $\eta_{A_{4}A_{5}}P_{4}^{M}$ term from
$\\{X_{45}^{M}\\}$ with the second term inside the second curly bracket of
(3.31) is (similar to (3.20)), :

$\displaystyle\frac{2}{h+c-1}\eta_{A_{4}A_{5}}(d-1-\gamma_{45})\mathscr{H}_{A_{1}A_{2}A_{3}}(P_{1},P_{2},P_{3})\
,$ (3.32)

where we have used that
$\sum_{i=1}^{3}\gamma_{i4}+\gamma_{45}=\Delta_{4}=d-1$. Since (3.32) is
symmetric under $4\leftrightarrow 5$, its contribution vanishes due to the
antisymmetric property of $\\{X_{45,M}\\}$. The product of
$\eta_{A_{4}M}P_{4,A_{5}}$ in $\\{X_{45}^{M}\\}$, the second term inside the
second curly bracket of (3.31) and

$\displaystyle
f(\gamma_{45})=\frac{\Gamma\left(\gamma_{45}-\frac{\Delta_{4}+\Delta_{5}-h+c+1}{2}\right)}{\Gamma(\gamma_{45})}\
,$ (3.33a) similar to (3.21b), gives
$f(\gamma_{45})\left(P_{1,A_{4}}+P_{2,A_{4}}+P_{3,A_{4}}\right)P_{4,A_{5}}\mathscr{H}_{A_{1}A_{2}A_{3}}(P_{1},P_{2},P_{3})=\\\
\left(f(\gamma_{45})\frac{1}{2}\eta_{A_{4}A_{5}}-f(\gamma_{45}-1)P_{4,A_{5}}P_{5,A_{4}}\right)\mathscr{H}_{A_{1}A_{2}A_{3}}(P_{1},P_{2},P_{3})+\cdots\
,$ (3.33b)

where $\cdots$ denotes the term vanishing upon the action of
$\widehat{D}^{M_{4}A_{4}}$. The expression (3.33b) exhibits symmetry under
$4\leftrightarrow 5$. As a result, again due to the antisymmetry of
$\\{X_{45}^{M}\\}$, (3.33b) gets canceled out. Consequently, based on the
preceding arguments, we can conclude that the $\mathscr{H}_{A_{1}A_{2}A_{3}}$
term of (3.31) can be safely disregarded. Then, the same as in the other
channel, the poles of $c$ become simple. And we can integrate around the pole
$c=h-1$.

Besides, the pole of $\gamma_{ij}^{\prime}$ (for $1\leq i<j\leq 3$) is at
$\gamma_{ij}+n_{ij}$, with any non-negative integers $n_{ij}$. Integrating
over it around the pole, we have

$\int\prod_{k<\ell}^{3}\frac{d\gamma_{k\ell}^{\prime}}{2\pi i}\
\delta\left(\sum_{k<\ell}^{3}\gamma_{k\ell}^{\prime}-d\right)\prod_{i<j}^{3}\frac{\Gamma\left(\gamma_{ij}-\gamma_{ij}^{\prime}\right)\Gamma(\gamma_{ij}^{\prime})}{\Gamma(\gamma_{ij})}\frac{1}{\gamma_{12}^{\prime}-\frac{d}{2}+n}\\\
=\sum_{m=0}^{\infty}\frac{1}{\sum_{i<j}^{3}\gamma_{ij}-d+m}\sum_{\sum_{i<j}^{3}n_{ij}=m}(-1)^{m}\frac{(\gamma_{12})_{n_{12}}(\gamma_{13})_{n_{13}}(\gamma_{23})_{n_{23}}}{n_{12}!n_{13}!n_{23}!}\frac{1}{\gamma_{12}+n_{12}-\frac{d}{2}+n}\\\
=\sum_{m=0}^{\infty}\frac{1}{\sum_{i<j}^{3}\gamma_{ij}-d+m}\sum_{n_{12}=0}^{m}(-1)^{m}\frac{(\gamma_{12})_{n_{12}}(\gamma_{13}+\gamma_{23})_{m-n_{12}}}{n_{12}!(m-n_{12})!}\frac{1}{\gamma_{12}+n_{12}-\frac{d}{2}+n}\
.$ (3.34a) The above equation has poles at
$\sum_{i<j}^{3}\gamma_{ij}=d-m\quad\text{and}\quad\gamma_{12}=\frac{d}{2}-n-n_{12}.$
(3.34b) Using (3.34b) and shifting $n\to n-n_{12}$, we get
$\sum_{n_{12}=0}^{m}\frac{(\gamma_{12})_{n_{12}}(\gamma_{13}+\gamma_{23})_{m-n_{12}}}{n_{12}!(m-n_{12})!}\frac{\left(\Gamma(d-1)(\frac{d}{2}-n)_{n}\right)^{2}}{4n!\Gamma\left(\frac{d}{2}+n\right)\left(\gamma_{12}+n_{12}-\frac{d}{2}+n\right)}\\\
\to{\mathbf{{V}}}_{3}^{m,n,0}\frac{\Gamma(d-1)(\frac{d}{2}-n)_{n}}{4m!n!\Gamma\left(\frac{d}{2}+n\right)\left(\gamma_{12}-\frac{d}{2}+n\right)}\
,$ (3.34c) where $\displaystyle{\mathbf{{V}}}_{3}^{m,n,0}\equiv$
$\displaystyle\Gamma(d-1)m!\sum_{n_{12}=0}^{\min\\{m,n\\}}\frac{(\frac{d}{2}-m+n)_{m}(\frac{d}{2}-n+n_{12})_{n-n_{12}}(n-n_{12}+1)_{n_{12}}}{n_{12}!(m-n_{12})!}$
(3.34d) $\displaystyle=$
$\displaystyle\Gamma(d-1)\left(\frac{d}{2}-n+m\right)_{n}\left(\frac{d}{2}-m+n\right)_{m}\
.$

Note that ${\mathbf{{V}}}_{3}^{m,n,0}$ is explicitly symmetric under
$m\leftrightarrow n$. It can also be easily checked that with $n=0$,
${\mathbf{{V}}}_{3}^{m,0,0}$ reduces to (3.14). Now, we can rewrite the Mellin
amplitude in the simplified form

$\mathscr{M}^{M_{1}\cdots
M_{5}}_{\texttt{3v3v3v}}=-ig^{3}f^{a_{1}a_{2}b^{\prime}}f^{a_{3}bb^{\prime}}f^{ba_{4}a_{5}}\frac{\pi^{h}}{2}\left(\prod_{i=1}^{5}\mathfrak{D}^{M_{i}A_{i}}\right)\sum_{m,n=0}^{\infty}\Big{\\{}X_{bb^{\prime},A_{3}}\Big{\\}}{\mathbf{{V}}}_{3}^{m,n,0}\\\
\times\frac{\Big{\\{}X_{45}^{A_{b}}\Big{\\}}{\mathbf{{V}}}_{3}^{m,0,0}}{4m!\Gamma\left(\frac{d}{2}+m\right)\left(\gamma_{45}-\frac{d}{2}+m\right)}\times\frac{\Big{\\{}X_{12}^{A_{b^{\prime}}}\Big{\\}}{\mathbf{{V}}}_{3}^{n,0,0}}{4n!\Gamma\left(\frac{d}{2}+n\right)\left(\gamma_{12}-\frac{d}{2}+n\right)}\
.$ (3.35)

For even $d$, the infinite sums are cut-off at $m=\frac{d}{2}$ and
$n=\frac{d}{2}$, and only finite number of terms remain.

Just as in the channel (3.25), we can compare (3.35) to its flat-space
counterpart,

$\mathscr{A}_{\texttt{3v3v3v}}=-g^{3}f^{a_{1}a_{2}b^{\prime}}f^{a_{3}bb^{\prime}}f^{ba_{4}a_{5}}\left(\eta_{MM^{\prime}}q_{A_{3}}-2\eta_{MA_{3}}q_{M^{\prime}}-(qM\leftrightarrow
q^{\prime}M^{\prime})\right)\\\
\times\left(\eta_{A_{4}A_{5}}k_{4}^{M}-2\eta_{A_{4}}^{M}k_{4,A_{5}}-(4\leftrightarrow
5)\right)\frac{i}{(k_{4}+k_{5})^{2}}\\\
\times\left(\eta_{A_{1}A_{2}}k_{1}^{M^{\prime}}-2\eta_{A_{1}}^{M^{\prime}}k_{1,A_{2}}-(1\leftrightarrow
2)\right)\frac{i}{(k_{1}+k_{2})^{2}}\ ,$ (3.36)

with $q\equiv-k_{1}-k_{2}-k_{3}$ and $q^{\prime}\equiv k_{1}+k_{2}$. The
dictionary between the Mellin amplitude and the flat-space amplitude, which is
obtained from the Feynman rules, can be read off as follows. For the momenta,
$ik_{i}\to 2P_{i}$. For the propagator

$\frac{i}{(k_{4}+k_{5})^{2}}\to\frac{1}{4m!\Gamma\left(\frac{d}{2}+m\right)\left(\gamma_{45}-\frac{d}{2}+m\right)}\
,$ (3.37)

with $m$ to be summed over. And

$\frac{i}{(k_{1}+k_{2})^{2}}\to\frac{1}{4n!\Gamma\left(\frac{d}{2}+n\right)\left(\gamma_{12}-\frac{d}{2}+n\right)}\
,$ (3.38)

with $n$ to be summed over. For the three-vertex coupling constant,

$\displaystyle g\to g\ {\mathbf{{V}}}_{3}^{m,0,0},\quad g\
{\mathbf{{V}}}_{3}^{n,0,0},\quad g\ {\mathbf{{V}}}_{3}^{m,n,0}\ .$ (3.39)

Note that the former two, each of which has two zero arguments, are for the
three-vertices connected to two external fields, while the last one, which has
only one zero argument, is for the three-vertex connected to only one external
fields.

##  4 Higher-point gluon amplitudes

###  4.1 Factorization of $(n+1)$-point gluon amplitudes

Figure 4:  A $(n+1)$-current amplitude for involving a three-vertex

So far we have presented lower-point gluon amplitudes (three to five-point)
that already exist in the literature. Particularly, we have considerably
simplified the five-point result. There we demonstrated how the calculation of
the five-point gluon amplitude can be done by factorizing it into a four-point
amplitude and a three-point amplitude. One interesting finding is that the
antisymmetry between the two external legs $P_{4}$ and $P_{5}$, which are
brought together to a three-vertex, kills the term
$\mathscr{H}_{A_{1}A_{2}A_{3}}$. And the remaining term has simple poles at
$c=\pm(h-1)$.

It is important to observe that this simplification does not depend on the
details of the four-point amplitude, i.e. $\mathscr{G}_{A_{1}A_{2}A_{3}M}$ and
$\mathscr{H}_{A_{1}A_{2}A_{3}}$ in (3.30b). So, presumably for any
$(n+1)$-point gluon amplitude factorized into an $n$-point amplitude and a
three-vertex (see Figure 4), we expect a similar simplification. In this
section we will explicitly demonstrate this fact and apply it in higher-point
computations.

First, from the factorization of bulk-to-bulk propagator (2.15), we can
calculate the $(n+1)$-point gluon amplitude from the lower-point amplitudes.

$\left\langle\prod_{i=1}^{n+1}J^{M_{i}}(P_{i})\right\rangle=\int_{-i\infty}^{i\infty}\frac{dc}{2\pi
i}f_{\Delta}(c)\int_{\partial\text{AdS}}dQ\left\langle\prod_{i=1}^{n-1}J^{M_{i}}(P_{i})J^{M}_{h+c}(Q)\right\rangle\\\
\times\eta_{MN}\left\langle
J^{N}_{h-c}(Q)J^{M_{n}}(P_{n})J^{M_{n+1}}(P_{n+1})\right\rangle\ .$ (4.1)

An $n$-point Mellin amplitude can be written in the following form.

$\mathscr{M}^{M_{1}M_{2}\cdots
M_{n}}=\frac{\pi^{h}}{2}\left(\prod_{i=1}^{n}\mathfrak{D}^{M_{i}A_{i}}\right)\tilde{\mathscr{M}}^{a_{1}a_{2}\cdots
a_{n}}_{A_{1}A_{2}\cdots
A_{n}}=\frac{\pi^{h}}{2}\left(\prod_{i=1}^{n-1}\mathfrak{D}^{M_{i}A_{i}}\right)\frac{C_{n}}{\Gamma(\Delta_{n})}\\\
\left(\frac{\Delta_{n}-1}{\Delta_{n}}\eta^{M_{n}A_{n}}\tilde{\mathscr{M}}^{a_{1}a_{2}\cdots
a_{n}}_{A_{1}A_{2}\cdots A_{n}}+\frac{1}{\Delta_{n}}\frac{\partial}{\partial
P_{n}^{M_{n}}}\mathscr{H}^{a_{1}a_{2}\cdots a_{n}}_{A_{1}A_{2}\cdots
A_{n-1}}\right)\ ,$ (4.2)

where $\mathscr{H}^{a_{1}a_{2}\cdots a_{n}}_{A_{1}A_{2}\cdots A_{n-1}}\equiv
P_{n}^{A_{n}}\tilde{\mathscr{M}}^{a_{1}a_{2}\cdots a_{n}}_{A_{1}A_{2}\cdots
A_{n}}$.

As in the examples (2.11), (2.12) and their generalizations with more free
indices, we can always use (2.10) to replace $P_{n}^{A}$ by the other
$P_{i}^{A}$’s. In this way, we can eliminate the dependences of
$\tilde{\mathscr{M}}_{A_{1}A_{2}\cdots A_{n}}$ and
$\mathscr{H}_{A_{1}A_{2}\cdots A_{n-1}}$ on $P_{n}$. Then, the second term in
the last line of (4.2) can be further calculated as follows.

$\frac{\partial}{\partial P_{n}^{M_{n}}}\mathscr{H}_{A_{1}A_{2}\cdots
A_{n-1}}(P_{1},P_{2},\cdots,P_{n-1})\prod_{k<\ell}^{n}\Gamma(\gamma_{k\ell}^{\prime})P_{k\ell}^{-\gamma_{k\ell}^{\prime}}=\\\
2\sum_{i=1}^{n-1}P_{i}^{M_{n}}\mathscr{H}_{A_{1}A_{2}\cdots
A_{n-1}}(\gamma_{in}^{\prime}\to\gamma_{in}^{\prime}-1)\prod_{k<\ell}^{n}\Gamma(\gamma_{k\ell}^{\prime})P_{k\ell}^{-\gamma_{k\ell}^{\prime}}\
,$ (4.3)

where $\gamma_{ij}^{\prime}$ denote the Mellin variables for the lower point
Mellin amplitude $\mathscr{M}_{n}$. Plugging (3.11), (4.2) and (4.3) in (4.1),
we have

$\mathscr{M}^{M_{1}\cdots
M_{n}}=igf^{a_{n}a_{n+1}b}\frac{\pi^{h}}{2}\int_{-i\infty}^{i\infty}\frac{dc}{2\pi
i}\frac{c^{2}-(h-1)^{2}}{\left(c^{2}-(\Delta-h)^{2}\right)^{2}}\left(\prod_{i=1}^{n+1}\mathfrak{D}^{M_{i}A_{i}}\right)\frac{1}{2\Gamma(c)\Gamma(-c)}\\\
\times\Gamma\left(\frac{\Delta_{n}+\Delta_{n+1}+h-c-d+1}{2}\right)\Gamma\left(\frac{\Delta_{n}+\Delta_{n+1}-h+c+1}{2}\right)\frac{\Gamma\left(\gamma_{n(n+1)}-\frac{\Delta_{n}+\Delta_{n+1}-h+c+1}{2}\right)}{\Gamma(\gamma_{n(n+1)})}\\\
\times\int\prod_{i<j}^{n-1}\frac{d\gamma_{ij}^{\prime}}{2\pi
i}\frac{\Gamma\left(\gamma_{ij}-\gamma_{ij}^{\prime}\right)\Gamma(\gamma_{ij}^{\prime})}{\Gamma(\gamma_{ij})}\delta\left(\sum_{k<\ell}^{n-1}\gamma_{k\ell}^{\prime}+\frac{h+c-\sum_{i=1}^{n-1}\Delta_{i}^{\prime}}{2}\right)\\\
\times\Big{\\{}X_{n(n+1)}^{M}\Big{\\}}\left\\{\tilde{\mathscr{M}}_{A_{1}A_{2}\cdots
A_{n-1}M}^{a_{1}a_{2}\cdots
a_{n-1}b}(\gamma_{ij}^{\prime})+\frac{2}{h+c-1}\sum_{i=1}^{n-1}P_{i,M}\mathscr{H}^{a_{1}a_{2}\cdots
a_{n-1}b}_{A_{1}A_{2}\cdots A_{n-1}}(\gamma_{ij}^{\prime})\right\\}\ .$ (4.4)

Again $\Delta_{i}^{\prime}\equiv\Delta_{i}-\delta_{i}$ where $\delta_{i}$
denotes the scaling dimension of $P_{i}$ for each term in the last line. Note
that for all the terms in the sum in the last line, they have the same total
scaling dimension $\sum_{i}^{n-1}\Delta_{i}^{\prime}$.

Now we take $\Delta_{i}=d-1$ and show that the contribution of
$\mathscr{H}_{A_{1}A_{2}\cdots A_{n-1}}$ vanishes. First,

$\eta_{A_{n}A_{n+1}}P_{n}^{M}\sum_{i=1}^{n-1}P_{i,M}\mathscr{H}_{A_{1}A_{2}\cdots
A_{n-1}}\to-\frac{1}{2}\eta_{A_{n}A_{n+1}}(d-1-\gamma_{n(n+1)})\mathscr{H}_{A_{1}A_{2}\cdots
A_{n-1}}\ ,$ (4.5)

where we have shifted the Mellin variables $\gamma_{in}\to\gamma_{in}+1$ (no
prime) for each term in the sum and used that
$\sum_{i=1}^{n-1}\gamma_{in}+\gamma_{n(n+1)}=\Delta_{n}=d-1$. Since (4.5) is
symmetric under the interchange of labels $n$ and $n+1$ (i.e.,
$n\leftrightarrow n+1$), the contribution of it vanishes due to the
antisymmetry of $\\{X_{n(n+1}^{M}\\}$. Second, using (2.12) with

$\displaystyle
f(\gamma_{n(n+1)})=\frac{\Gamma\left(\gamma_{n(n+1)}-\frac{\Delta_{n}+\Delta_{n+1}-h+c+1}{2}\right)}{\Gamma(\gamma_{n(n+1)})}\
,$ (4.6a) we can get
$f(\gamma_{n(n+1)})\sum_{i=1}^{n-1}P_{i,A_{n}}P_{n,A_{n+1}}\mathscr{H}_{A_{1}A_{2}\cdots
A_{n-1}}\\\
=\left(\frac{1}{2}f(\gamma_{n(n+1)})\eta_{A_{n}A_{n+1}}-f(\gamma_{n(n+1)}-1)P_{n,A_{n+1}}P_{n+1,A_{n}}\right)\mathscr{H}_{A_{1}A_{2}\cdots
A_{n-1}}+\cdots\ ,$ (4.6b)

where $\cdots$ denotes the term vanishing upon the action of
$\widehat{D}^{M_{n}A_{n}}$. The expression (4.6b) again exhibits symmetry
under $n\leftrightarrow n+1$. As a result, its contribution gets completely
canceled out from the antisymmetry of $\\{X_{n(n+1}^{M}\\}$. Combining these,
we can conclude that the $\mathscr{H}$ term in (4.4) vanishes.

Finally, the integration around the simple pole $c=\Delta-h$ (with
$\Delta=d-1$) and the poles $\gamma_{ij}^{\prime}=\gamma_{ij}+n_{ij}$ with any
non-negative integers $n_{ij}$ gives the important recursion formula from an
$n$-point amplitude to an $(n+1)$-point amplitude.

$\mathscr{M}^{M_{1}M_{2}\cdots
M_{n+1}}_{n+1}=\frac{\pi^{h}}{2}\left(\prod_{i=1}^{n+1}\mathfrak{D}^{M_{i}A_{i}}\right)igf^{a_{n}a_{n+1}b}\sum_{m=0}^{\infty}\frac{\Big{\\{}X_{n(n+1)}^{M}\Big{\\}}{\mathbf{{V}}}_{3}^{m,0,0}}{4\Gamma\left(\frac{d}{2}+m\right)(\gamma_{n(n+1)}-\frac{d}{2}+m)}\\\
\times\sum_{\sum_{i<j}^{n-1}n_{ij}=m}\prod_{i<j}^{n-1}\frac{(\gamma_{ij})_{n_{ij}}}{n_{ij}!}\tilde{\mathscr{M}}_{n,A_{1}A_{2}\cdots
A_{n-1}M}^{a_{1}a_{2}\cdots
a_{n-1}b}(P_{1},P_{2},\cdots,P_{n-1})\big{|}_{\gamma_{ij}^{\prime}\to\gamma_{ij}+n_{ij}}\
.$ (4.7)

One can use this recursion formula to calculate any $n$-point gluon amplitudes
with at most one four-vertex. For example, if a channel contains one four-
vertex, we can start from the four-vertex and add three-vertices one by one.
And at each step of adding one more three-vertex, the amplitude can be
calculated by using (4.7).

###  4.2 Six-point amplitude: snowflake channel

Figure 5:  The “snowflake” channel of six point gluon amplitude for (4.8)

We can use (4.7) to calculate the six-point gluon Mellin amplitude for the
diagram in Figure 5. As shown in the figure, the amplitude can be factorized
into a five-point gluon amplitude and a three-point gluon amplitude.
Therefore,

$\mathscr{M}_{\texttt{Snowflake}}^{M_{1}M_{2}\cdots
M_{6}}=\frac{\pi^{h}}{2}\left(\prod_{i=1}^{6}\mathfrak{D}^{M_{i}A_{i}}\right)igf^{a_{5}a_{6}b}\Big{\\{}X_{56}^{M}\Big{\\}}\sum_{m=0}^{\infty}\frac{{\mathbf{{V}}}_{3}^{m,0,0}}{4\Gamma\left(\frac{d}{2}+m\right)(\gamma_{56}-\frac{d}{2}+m)}\\\
\times\sum_{\sum_{i<j}^{4}n_{ij}=m}\prod_{i<j}^{4}\frac{(\gamma_{ij})_{n_{ij}}}{n_{ij}!}\tilde{\mathscr{M}}_{\texttt{3v3v3v},A_{1}A_{2}\cdots
A_{4}M}^{a_{1}a_{2}a_{3}a_{4}b}(P_{1},P_{2},P_{3},P_{4})\big{|}_{\gamma_{ij}^{\prime}\to\gamma_{ij}+n_{ij}}\
.$ (4.8)

In (4.8) $\tilde{\mathscr{M}}_{\texttt{3v3v3v}}$ can achieved by eliminating
the $P_{3}$ dependence in the Mellin amplitude (3.35), interchanging the
labels $1\leftrightarrow 2$ and $3\leftrightarrow 5$, and replacing
$(A_{5},a_{5})$ by $(M,b)$. We first eliminate $P_{3}$ in the Mellin
amplitude. One appearance of $P_{3}$ is either in the inner products
$P_{i}\cdot P_{3}-(i\leftrightarrow j)$ (where $(i,j)=(1,2)$ or $(4,5)$), or
in the terms with free indices, $P_{i,A_{j}}P_{3,A_{i}}-(i\leftrightarrow j)$.
For the former, by shifting the Mellin variable
$\gamma_{i(j)3}\to\gamma_{i(j)3}+1$ we have

$\displaystyle P_{i}\cdot P_{3}-(i\leftrightarrow
j)\to-\frac{1}{2}\gamma_{i3}-(i\leftrightarrow j)=\frac{1}{2}\sum_{k\neq
i,3}\gamma_{ik}-(i\leftrightarrow j)\to-\sum_{k\neq i}P_{i}\cdot
P_{k}-(i\leftrightarrow j)\ ,$ (4.9)

where in the second step we have used $\Delta_{i}=\Delta_{j}=d-1$ and in the
last step we have shifted $\gamma_{i(j)k}\to\gamma_{i(j)k}-1$. For the latter,
we can use (2.12) to replace $P_{3,A_{i(j)}}$. So, it becomes

$\displaystyle P_{i,A_{j}}P_{3,A_{i}}-(i\leftrightarrow j)\to-
P_{i,A_{j}}\sum_{k\neq 3}P_{k,A_{i}}-(i\leftrightarrow j)\ .$ (4.10)

Combining (4.9) and (4.10), the net result is that we can replace $P_{3}$ in
(3.35) by $-\sum_{k\neq 3}P_{k}$. After getting rid of the $P_{3}$ dependence,
and interchanging the labels $1\leftrightarrow 2$ and $3\leftrightarrow 5$, we
can read off that

$\tilde{\mathscr{M}}_{\texttt{3v3v3v},A_{1}A_{2}\cdots
A_{5}}^{a_{1}a_{2}\cdots
a_{5}}=-ig^{3}f^{a_{1}a_{2}b^{\prime}}f^{a_{5}b^{\prime\prime}b^{\prime}}f^{a_{3}a_{4}b^{\prime\prime}}\sum_{m^{\prime},n=0}^{\infty}\Big{\\{}X_{b^{\prime\prime}b^{\prime},A_{5}}\Big{\\}}{\mathbf{{V}}}_{3}^{m^{\prime},n,0}\\\
\times\frac{\Big{\\{}X_{34}^{A_{b^{\prime\prime}}}\Big{\\}}{\mathbf{{V}}}_{3}^{m^{\prime},0,0}}{4m^{\prime}!\Gamma\left(\frac{d}{2}+m^{\prime}\right)\left(\gamma_{34}^{\prime}-\frac{d}{2}+m^{\prime}\right)}\times\frac{\Big{\\{}X_{12}^{A_{b^{\prime}}}\Big{\\}}{\mathbf{{V}}}_{3}^{n,0,0}}{4n!\Gamma\left(\frac{d}{2}+n\right)\left(\gamma_{12}^{\prime}-\frac{d}{2}+n\right)}\
,$ (4.11)

with $P_{b^{\prime\prime}}\equiv P_{3}+P_{4}\ ,\ P_{b^{\prime}}\equiv
P_{1}+P_{2}$. Plugging it in (4.8), we see that the poles are at

$\gamma_{56}=\frac{d}{2}-m,\quad\gamma_{12}=\frac{d}{2}-n-n_{12},\quad\gamma_{34}=\frac{d}{2}-m^{\prime}-n_{34}.$
(4.12a) And from the delta function restriction,
$\displaystyle\gamma_{13}+\gamma_{14}+\gamma_{23}+\gamma_{24}=$
$\displaystyle\frac{\sum_{i=1}^{4}\Delta_{i}-\Delta_{5}-\Delta_{6}+2}{2}+\gamma_{56}-\gamma_{12}-\gamma_{34}$
(4.12b) $\displaystyle=$
$\displaystyle\frac{d}{2}-m+n+m^{\prime}+n_{12}+n_{34}\ .$ The sum over
$n_{ij}$ in (4.8) at the poles yields
$\displaystyle\sum_{\sum_{i<j}^{4}n_{ij}=m}\prod_{i<j}^{4}\frac{(\gamma_{ij})_{n_{ij}}}{n_{ij}!}=$
$\displaystyle\sum_{n_{12}=0}^{m}\sum_{n_{34}=0}^{m-n_{12}}\frac{(\frac{d}{2}-n-n_{12})_{n_{12}}}{n_{12}!}\frac{(\frac{d}{2}-m^{\prime}-n_{34})_{n_{34}}}{n_{34}!}$
(4.12c)
$\displaystyle\times\frac{(\frac{d}{2}-m+n+m^{\prime}+n_{12}+n_{34})_{m-n_{12}-n_{34}}}{(m-n_{12}-n_{34})!}\
.$ Shifts of $m^{\prime}\to m^{\prime}-n_{34}$ and $n\to n-n_{12}$ lead to
$\sum_{n_{12}=0}^{m}\sum_{n_{34}=0}^{m-n_{12}}\frac{(\frac{d}{2}-n-n_{12})_{n_{12}}}{n_{12}!}\frac{(\frac{d}{2}-m^{\prime}-n_{34})_{n_{34}}}{n_{34}!}\frac{(\frac{d}{2}-m+n+m^{\prime}+n_{12}+n_{34})_{m-n_{12}-n_{34}}}{(m-n_{12}-n_{34})!}\\\
\times\frac{\Gamma(d-1)(\frac{d}{2}-m^{\prime})_{m^{\prime}}}{4m^{\prime}!\Gamma\left(\frac{d}{2}+m^{\prime}\right)\left(\gamma_{34}+n_{34}-\frac{d}{2}+m^{\prime}\right)}{\mathbf{{V}}}_{3}^{m^{\prime},n,0}\frac{\Gamma(d-1)(\frac{d}{2}-n)_{n}}{4n!\Gamma\left(\frac{d}{2}+n\right)\left(\gamma_{12}+n_{12}-\frac{d}{2}+n\right)}\\\
\to\frac{1}{m!}\frac{\Gamma(d-1)(\frac{d}{2}-m^{\prime})_{m^{\prime}}}{4m^{\prime}!\Gamma\left(\frac{d}{2}+m^{\prime}\right)\left(\gamma_{34}-\frac{d}{2}+m^{\prime}\right)}{\mathbf{{V}}}_{3}^{m^{\prime},n,m}\frac{\Gamma(d-1)(\frac{d}{2}-n)_{n}}{4n!\Gamma\left(\frac{d}{2}+n\right)\left(\gamma_{12}-\frac{d}{2}+n\right)}\
,$ (4.12d) where we have defined
${\mathbf{{V}}}_{3}^{m^{\prime},n,m}\equiv\sum_{n_{12}=0}^{\min\\{m,n\\}}\sum_{n_{34}=0}^{\min\\{m-n_{12},m^{\prime}\\}}\frac{m!}{n_{12}!n_{34}!}\frac{\left(\frac{d}{2}-m+n+m^{\prime}\right)_{m-n_{12}-n_{34}}}{(m-n_{12}-n_{34})!}(m^{\prime}-n_{34}+1)_{n_{34}}\\\
\times\left(\frac{d}{2}+m^{\prime}-n_{34}\right)_{n_{34}}(n-n_{12}+1)_{n_{12}}\left(\frac{d}{2}+n-n_{12}\right)_{n_{12}}{\mathbf{{V}}}_{3}^{m^{\prime}-n_{34},n-n_{12},0}\
.$ (4.12e)

Presumably this definition is symmetric among $m^{\prime},n$ and $m$, and
reduces to the one defined in (3.34d) when one of the integer is set to 0.
While we don’t have a proof for the symmetry property, we can check that the
reduction property is true. First, plugging $m=0$ in (4.12e), we find that the
r.h.s reduces to ${\mathbf{{V}}}_{3}^{m^{\prime},n,0}$, consistent with the
l.h.s. Besides, from the definition (4.12e), we find
${\mathbf{{V}}}_{3}^{m^{\prime},0,m}={\mathbf{{V}}}_{3}^{0,m^{\prime},m}$, and
it is equal to ${\mathbf{{V}}}_{3}^{m,m^{\prime},0}$ from the definition in
(3.34d).

With the aid of the newly defined ${\mathbf{{V}}}_{3}^{m^{\prime},n,m}$, we
can rewrite the Mellin amplitude (4.8) as

$\mathscr{M}_{\texttt{Snowflake}}^{M_{1}M_{2}\cdots
M_{6}}=g^{4}f^{a_{1}a_{2}b^{\prime}}f^{a_{3}a_{4}b^{\prime\prime}}f^{a_{5}a_{6}b}f^{bb^{\prime\prime}b^{\prime}}\frac{\pi^{h}}{2}\left(\prod_{i=1}^{6}\mathfrak{D}^{M_{i}A_{i}}\right)\\\
\sum_{m^{\prime},n,m=0}^{\infty}\Big{\\{}X_{b^{\prime\prime}b^{\prime},M}\Big{\\}}{\mathbf{{V}}}_{3}^{m^{\prime},n,m}\times\frac{\Big{\\{}X_{56}^{M}\Big{\\}}{\mathbf{{V}}}_{3}^{m,0,0}}{4m!\Gamma\left(\frac{d}{2}+m\right)\left(\gamma_{56}-\frac{d}{2}+m\right)}\\\
\times\frac{\Big{\\{}X_{34}^{A_{b^{\prime\prime}}}\Big{\\}}{\mathbf{{V}}}_{3}^{m^{\prime},0,0}}{4m^{\prime}!\Gamma\left(\frac{d}{2}+m^{\prime}\right)\left(\gamma_{34}-\frac{d}{2}+m^{\prime}\right)}\times\frac{\Big{\\{}X_{12}^{A_{b^{\prime}}}\Big{\\}}{\mathbf{{V}}}_{3}^{n,0,0}}{4n!\Gamma\left(\frac{d}{2}+n\right)\left(\gamma_{12}-\frac{d}{2}+n\right)}\
.$ (4.13)

Compare it with the flat-space analog

$\mathscr{A}_{\texttt{Snowflake}}^{M_{1}M_{2}\cdots
M_{6}}=g^{4}f^{a_{1}a_{2}b^{\prime}}f^{a_{3}a_{4}b^{\prime\prime}}f^{a_{5}a_{6}b}f^{bb^{\prime\prime}b^{\prime}}\left(\eta_{M^{\prime\prime}M^{\prime}}q^{\prime\prime}_{M}-2\eta_{M^{\prime\prime}M}q^{\prime\prime}_{M^{\prime}}-(q^{\prime\prime}M^{\prime\prime}\leftrightarrow
q^{\prime}M^{\prime})\right)\\\
\times\left(\eta_{A_{5}A_{6}}k_{5}^{M}-2\eta_{A_{5}}^{M}k_{5,A_{6}}-(5\leftrightarrow
6)\right)\frac{i}{(k_{5}+k_{6})^{2}}\left(\eta_{A_{3}A_{4}}k_{3}^{M^{\prime\prime}}-2\eta_{A_{3}}^{M^{\prime\prime}}k_{3,A_{4}}-(3\leftrightarrow
4)\right)\\\
\times\frac{i}{(k_{3}+k_{4})^{2}}\left(\eta_{A_{1}A_{2}}k_{1}^{M^{\prime}}-2\eta_{A_{1}}^{M^{\prime}}k_{1,A_{2}}-(1\leftrightarrow
2)\right)\frac{i}{(k_{1}+k_{2})^{2}}\ ,$ (4.14)

where $q^{\prime\prime}\equiv k_{3}+k_{4}$ and $q^{\prime}\equiv k_{1}+k_{2}$,
we have the dictionary that $ik_{i}\to 2P_{i}$ for the momenta,

$\frac{i}{(k_{i}+k_{j})^{2}}\to\frac{1}{4m!\Gamma\left(\frac{d}{2}+m\right)\left(\gamma_{ij}-\frac{d}{2}+m\right)}\
,$ (4.15)

with $m$ to be summed over for the propagators, and $g\to g\
{\mathbf{{V}}}_{3}^{m^{\prime},n,m}$ for the three-vertex coupling constant.

###  4.3 Six-point amplitude: channel with two three-vertices and a four-
vertex

Now we calculate the six-point gluon Mellin amplitude in another channel as
depicted in Figure 6. The amplitude can be factorized into a five-point
amplitude and a three-point amplitude.

Figure 6:  The channel of six point gluon amplitude for (4.16)

Using (3.11), (3.25) and (4.7), we have

$\mathscr{M}_{\texttt{3v3v4v}}^{M_{1}M_{2}\cdots
M_{6}}=ig^{4}\frac{\pi^{h}}{2}\left(\prod_{i=1}^{6}\mathfrak{D}^{M_{i}A_{i}}\right)\sum_{m=0}^{\infty}\frac{{\mathbf{{V}}}_{3}^{m,0,0}}{4\Gamma\left(\frac{d}{2}+m\right)(\gamma_{56}-\frac{d}{2}+m)}\\\
\times\sum_{\sum_{i<j}^{4}n_{ij}=m}\prod_{i<j}^{4}\frac{(\gamma_{ij})_{n_{ij}}}{n_{ij}!}\sum_{n=0}^{\infty}\frac{{\mathbf{{V}}}_{3}^{n,0,0}\times{\mathbf{{V}}}_{4}^{n,0,0,0}}{4n!\Gamma\left(\frac{d}{2}+n\right)(\gamma_{34}+n_{34}-\frac{d}{2}+n)}f^{a_{5}a_{6}b}\Big{\\{}X_{56}^{M}\Big{\\}}f^{a_{3}a_{4}b^{\prime\prime}}\\\
\times\left(\left(f^{a_{1}b^{\prime\prime}b^{\prime}}f^{a_{2}bb^{\prime}}+f^{a_{2}b^{\prime\prime}b^{\prime}}f^{a_{1}bb^{\prime}}\right)\Big{\\{}X_{34,M}\Big{\\}}\eta_{A_{1}A_{2}}+\text{cyclic
perm. of }(A_{1}a_{1},A_{2}a_{2},Mb)\right)\ .$ (4.16) The poles of Mellin
variables are at
$\displaystyle\gamma_{56}=\frac{d}{2}-m,\quad\gamma_{34}=\frac{d}{2}-n-n_{34}\
.$ (4.17a) Then, from the equation which results from the restrictions on the
Mellin variables, i.e.,
$\displaystyle\Delta_{5}+\Delta_{6}+1-2\gamma_{56}=\sum_{i=1}^{4}\Delta_{i}+1-2\sum_{i<j}^{4}\gamma_{ij}\
,$ (4.17b) we have
$\displaystyle\sum_{\sum_{i<j}^{4}n_{ij}=m}\prod_{i<j}^{4}\frac{(\gamma_{ij})_{n_{ij}}}{n_{ij}!}=\sum_{n_{34}=0}^{m}\frac{(\frac{d}{2}-n-n_{34})_{n_{34}}(d-1-m+n+n_{34})_{m-n_{34}}}{n_{34}!(m-n_{34})!}\
.$ (4.17c) A shift of $n\to n-n_{34}$ amounts to
$\displaystyle\sum_{n_{34}=0}^{m}\frac{(\frac{d}{2}-n-n_{34})_{n_{34}}(d-1-m+n+n_{34})_{m-n_{34}}}{n_{34}!(m-n_{34})!}\frac{\left(\frac{d}{2}-n\right)_{n}(d-1-n)_{n}}{n!\Gamma\left(\frac{d}{2}+n\right)(\gamma_{34}+n_{34}-\frac{d}{2}+n)}$
(4.17d) $\displaystyle\to$
$\displaystyle\frac{{\mathbf{{V}}}_{4}^{m,n,0,0}}{\Gamma\left(\frac{3d-4}{2}\right)}\frac{\left(\frac{d}{2}-n\right)_{n}}{m!n!\Gamma\left(\frac{d}{2}+n\right)(\gamma_{34}-\frac{d}{2}+n)}\
,$ where $\displaystyle{\mathbf{{V}}}_{4}^{m,n,0,0}\equiv$
$\displaystyle\Gamma\left(\frac{3d-4}{2}\right)m!\sum_{n_{34}=0}^{\min\\{m,n\\}}\frac{(d-1-m+n)_{m-n_{34}}}{n_{34}!(m-n_{34})!}$
(4.17e)
$\displaystyle\times(n-n_{34}+1)_{n_{34}}\left(\frac{d}{2}+n-n_{34}\right)_{n_{34}}(d-1-n+n_{34})_{n-n_{34}}\
.$

One can check that from this definition
${\mathbf{{V}}}_{4}^{m,0,0,0}={\mathbf{{V}}}_{4}^{0,m,0,0}$, which is also
identical to the ${\mathbf{{V}}}_{4}^{m,0,0,0}$ given in (3.26). Now, we can
rewrite (4.16) as

$\mathscr{M}_{\texttt{3v3v4v}}^{M_{1}M_{2}\cdots
M_{6}}=ig^{4}\frac{\pi^{h}}{2}\left(\prod_{i=1}^{6}\mathfrak{D}^{M_{i}A_{i}}\right)\sum_{m,n=0}^{\infty}{\mathbf{{V}}}_{4}^{m,n,0,0}\times\frac{\Big{\\{}X_{56}^{M}\Big{\\}}{\mathbf{{V}}}_{3}^{m,0,0}}{4m!\Gamma\left(\frac{d}{2}+m\right)(\gamma_{56}-\frac{d}{2}+m)}\\\
\times\frac{\Big{\\{}X_{34}^{M^{\prime\prime}}\Big{\\}}{\mathbf{{V}}}_{3}^{n,0,0}}{4n!\Gamma\left(\frac{d}{2}+n\right)(\gamma_{34}-\frac{d}{2}+n)}f^{a_{5}a_{6}b}f^{a_{3}a_{4}b^{\prime\prime}}\left(\left(f^{a_{1}b^{\prime\prime}b^{\prime}}f^{a_{2}bb^{\prime}}+f^{a_{2}b^{\prime\prime}b^{\prime}}f^{a_{1}bb^{\prime}}\right)\right.\\\
\times\left.\eta_{A_{1}A_{2}}\eta_{MM^{\prime\prime}}+\text{cyclic perm. of
}(A_{1}a_{1},A_{2}a_{2},Mb)\right)\ .$ (4.18)

Like the previous cases, the Mellin amplitude (4.18) can be obtained from the
flat-space Feynman rules by replacing $ik_{i}\to 2P_{i}$ for the momenta,

$\frac{i}{(k_{i}+k_{j})^{2}}\to\frac{1}{4m!\Gamma\left(\frac{d}{2}+m\right)\left(\gamma_{ij}-\frac{d}{2}+m\right)}\
,$ (4.19)

with $m$ to be summed over for the propagators, $g\to g\
{\mathbf{{V}}}_{3}^{m,0,0}$ for the three-vertex coupling constant and
$g^{2}\to g^{2}\ {\mathbf{{V}}}_{4}^{m,n,0,0}$ for the four-vertex coupling
constant.

###  4.4 Seven-point amplitude: scarecrow channel

Figure 7:  The "scarecrow" channel of seven-point gluon amplitude for (4.20)

Let’s proceed to compute the seven-point gluon amplitude, as depicted in
Figure 7. Using (3.11), (4.18) and (4.7), we have

$\mathscr{M}_{\texttt{Scarecrow}}^{M_{1}M_{2}\cdots
M_{7}}=-g^{5}\frac{\pi^{h}}{2}\left(\prod_{i=1}^{7}\mathfrak{D}^{M_{i}A_{i}}\right)\sum_{m=0}^{\infty}\frac{\Gamma\left(d-1\right)\left(\frac{d}{2}-m\right)_{m}}{4\Gamma\left(\frac{d}{2}+m\right)(\gamma_{67}-\frac{d}{2}+m)}\\\
\times\sum_{\sum_{i<j}^{5}n_{ij}=m}\prod_{i<j}^{5}\frac{(\gamma_{ij})_{n_{ij}}}{n_{ij}!}\sum_{m^{\prime},n=0}^{\infty}{\mathbf{{V}}}_{4}^{m^{\prime},n,0,0}\frac{\Gamma\left(d-1\right)\left(\frac{d}{2}-m^{\prime}\right)_{m^{\prime}}}{4m^{\prime}!\Gamma\left(\frac{d}{2}+m^{\prime}\right)(\gamma_{45}+n_{45}-\frac{d}{2}+m^{\prime})}\\\
\times\frac{\Gamma\left(d-1\right)\left(\frac{d}{2}-n\right)_{n}}{4n!\Gamma\left(\frac{d}{2}+n\right)(\gamma_{23}+n_{23}-\frac{d}{2}+n)}f^{ba_{6}a_{7}}\Big{\\{}X_{67}^{M}\Big{\\}}f^{a_{4}a_{5}c}\Big{\\{}X_{45}^{N}\Big{\\}}f^{a_{2}a_{3}b^{\prime\prime}}\\\
\times\left(\left(f^{bb^{\prime\prime}b^{\prime}}f^{a_{1}cb^{\prime}}+f^{a_{1}b^{\prime\prime}b^{\prime}}f^{bcb^{\prime}}\right)\Big{\\{}X_{23,N}\Big{\\}}\eta_{MA_{1}}+\text{cyclic
perm. of }(Mb,A_{1}a_{1},Nc)\right)\ .$ (4.20)

At the pole,

$\displaystyle\gamma_{67}=\frac{d}{2}-m,\quad\gamma_{23}=\frac{d}{2}-n-n_{23},\quad\gamma_{45}=\frac{d}{2}-m^{\prime}-n_{45}\
,$ (4.21a) with
$\Delta_{6}+\Delta_{7}+1-2\gamma_{67}=\sum_{i=1}^{5}\Delta_{i}+2-2\sum_{i<j}^{5}\gamma_{ij}$,
we have
$\displaystyle\sum_{\sum_{i<j}^{5}n_{ij}=m}\prod_{i<j}^{5}\frac{(\gamma_{ij})_{n_{ij}}}{n_{ij}!}$
(4.21b) $\displaystyle=$
$\displaystyle\sum_{n_{23}=0}^{m}\sum_{n_{45}=0}^{m-n_{23}}\frac{(\frac{d}{2}-n-n_{23})_{n_{23}}(\frac{d}{2}-m^{\prime}-n_{45})_{n_{45}}(d-1-m+n+m^{\prime}+n_{23}+n_{45})_{m-n_{23}-n_{45}}}{n_{23}!n_{45}!(m-n_{23}-n_{45})!}\
.$ Shift $n\to n-n_{23}$ and $m^{\prime}\to m^{\prime}-n_{45}$
$\sum_{n_{23}=0}^{m}\sum_{n_{45}=0}^{m-n_{23}}\frac{(\frac{d}{2}-n-n_{23})_{n_{23}}(\frac{d}{2}-m^{\prime}-n_{45})_{n_{45}}(d-1-m+n+m^{\prime}+n_{23}+n_{45})_{m-n_{23}-n_{45}}}{n_{23}!n_{45}!(m-n_{23}-n_{45})!}\\\
\times\frac{\left(\frac{d}{2}-m^{\prime}\right)_{m^{\prime}}}{m^{\prime}!\Gamma\left(\frac{d}{2}+m^{\prime}\right)(\gamma_{45}+n_{45}-\frac{d}{2}+m^{\prime})}{\mathbf{{V}}}_{4}^{m^{\prime},n,0,0}\frac{\left(\frac{d}{2}-n\right)_{n}}{n!\Gamma\left(\frac{d}{2}+n\right)(\gamma_{23}+n_{23}-\frac{d}{2}+n)}\\\
\to{\mathbf{{V}}}_{4}^{m^{\prime},n,m,0}\frac{1}{m!}\frac{\left(\frac{d}{2}-m^{\prime}\right)_{m^{\prime}}}{m^{\prime}!\Gamma\left(\frac{d}{2}+m^{\prime}\right)(\gamma_{45}-\frac{d}{2}+m^{\prime})}\frac{\left(\frac{d}{2}-n\right)_{n}}{n!\Gamma\left(\frac{d}{2}+n\right)(\gamma_{23}-\frac{d}{2}+n)}\
,$ (4.21c) where ${\mathbf{{V}}}_{4}^{m^{\prime},n,m,0}\equiv
m!\sum_{n_{23}=0}^{\min\\{m,n\\}}\sum_{n_{45}=0}^{\min\\{m-n_{23},m^{\prime}\\}}\frac{(d-1-m+n+m^{\prime})_{m-n_{23}-n_{45}}}{n_{23}!n_{45}!(m-n_{23}-n_{45})!}(m^{\prime}-n_{45}+1)_{n_{45}}\\\
\times\left(\frac{d}{2}+m^{\prime}-n_{45}\right)_{n_{45}}{\mathbf{{V}}}_{4}^{m^{\prime}-n_{45},n-n_{23},0,0}\times(n-n_{23}+1)_{n_{23}}\left(\frac{d}{2}+n-n_{23}\right)_{n_{23}}\
.$ (4.21d)

One can check that from this definition
${\mathbf{{V}}}_{4}^{m,n,0,0}={\mathbf{{V}}}_{4}^{0,m,n,0}={\mathbf{{V}}}_{4}^{m,0,n,0}$
and reduces to ${\mathbf{{V}}}_{4}^{m,n,0,0}$ in (4.17e). Furthermore, with
(4.21d), we can rewrite the Mellin amplitude (4.20) as

$\mathscr{M}_{\texttt{Scarecrow}}^{M_{1}M_{2}\cdots
M_{7}}=-g^{5}\frac{\pi^{h}}{2}\left(\prod_{i=1}^{7}\mathfrak{D}^{M_{i}A_{i}}\right)\sum_{m,n,m^{\prime}=0}^{\infty}{\mathbf{{V}}}_{4}^{m^{\prime},n,m,0}\frac{\Big{\\{}X_{67}^{M}\Big{\\}}{\mathbf{{V}}}_{3}^{m,0,0}}{4m!\Gamma\left(\frac{d}{2}+m\right)(\gamma_{67}-\frac{d}{2}+m)}\\\
\times\frac{\Big{\\{}X_{45}^{M^{\prime}}\Big{\\}}{\mathbf{{V}}}_{3}^{m^{\prime},0,0}}{4m^{\prime}!\Gamma\left(\frac{d}{2}+m^{\prime}\right)(\gamma_{45}-\frac{d}{2}+m^{\prime})}\times\frac{\Big{\\{}X_{23}^{N}\Big{\\}}{\mathbf{{V}}}_{3}^{n,0,0}}{4n!\Gamma\left(\frac{d}{2}+n\right)(\gamma_{23}-\frac{d}{2}+n)}f^{a_{6}a_{7}b}f^{a_{4}a_{5}c}f^{a_{2}a_{3}b^{\prime\prime}}\\\
\times\left(\left(f^{bb^{\prime\prime}b^{\prime}}f^{a_{1}cb^{\prime}}+f^{a_{1}b^{\prime\prime}b^{\prime}}f^{bcb^{\prime}}\right)\eta_{MA_{1}}\eta_{M^{\prime}N}+\text{cyclic
perm. of }(Mb,A_{1}a_{1},M^{\prime}c)\right)\ ,$ (4.22)

which can be mapped from the flat-space counterpart by replacing $ik_{i}\to
2P_{i}$ for the momenta,

$\frac{i}{(k_{i}+k_{j})^{2}}\to\frac{1}{4m!\Gamma\left(\frac{d}{2}+m\right)\left(\gamma_{ij}-\frac{d}{2}+m\right)}\
,$ (4.23)

with $m$ to be summed over for the propagators, $g\to g\
{\mathbf{{V}}}_{3}^{m,0,0}$ for the three-vertex coupling constant and
$g^{2}\to g^{2}\ {\mathbf{{V}}}_{4}^{m^{\prime},n,m,0}$ for the four-vertex
coupling constant.

###  4.5 Eight-point amplitude: drone channel

Figure 8:  The “drone” channel of eight-point gluon amplitude for (4.24)

In this subsection we calculate the eight-point gluon amplitude in the drone
channel. As shown in Figure 8. To calculate the amplitude, we factorize the
diagram into a seven-point amplitude and a three-point amplitude. Using
(3.11), (4.22) and (4.7), we have

$\mathscr{M}_{\texttt{Drone}}^{M_{1}M_{2}\cdots
M_{8}}=-ig^{6}\frac{\pi^{h}}{2}\left(\prod_{i=1}^{8}\mathfrak{D}^{M_{i}A_{i}}\right)\sum_{m=0}^{\infty}\frac{\Gamma\left(d-1\right)\left(\frac{d}{2}-m\right)_{m}}{4m!\Gamma\left(\frac{d}{2}+m\right)(\gamma_{78}-\frac{d}{2}+m)}\sum_{\sum_{i<j}^{6}n_{ij}=m}\prod_{i<j}^{6}\frac{(\gamma_{ij})_{n_{ij}}}{n_{ij}!}\\\
\sum_{n^{\prime},n,m^{\prime}=0}^{\infty}{\mathbf{{V}}}_{4}^{m^{\prime},n,n^{\prime},0}\frac{\Gamma\left(d-1\right)\left(\frac{d}{2}-n^{\prime}\right)_{n^{\prime}}}{4n^{\prime}!\Gamma\left(\frac{d}{2}+n^{\prime}\right)(\gamma_{56}+n_{56}-\frac{d}{2}+n^{\prime})}\frac{\Gamma\left(d-1\right)\left(\frac{d}{2}-m^{\prime}\right)_{m^{\prime}}}{4m^{\prime}!\Gamma\left(\frac{d}{2}+m^{\prime}\right)(\gamma_{34}+n_{34}-\frac{d}{2}+m^{\prime})}\\\
\times\frac{\Gamma\left(d-1\right)\left(\frac{d}{2}-n\right)_{n}}{4n!\Gamma\left(\frac{d}{2}+n\right)(\gamma_{12}+n_{12}-\frac{d}{2}+n)}f^{a_{7}a_{8}b}\Big{\\{}X_{78}^{M}\Big{\\}}f^{a_{5}a_{6}c^{\prime}}\Big{\\{}X_{56}^{M^{\prime}}\Big{\\}}f^{a_{3}a_{4}c}\Big{\\{}X_{34}^{N}\Big{\\}}\\\
\times\left(\left(f^{c^{\prime}b^{\prime\prime}b^{\prime}}f^{bcb^{\prime}}+f^{bb^{\prime\prime}b^{\prime}}f^{c^{\prime}cb^{\prime}}\right)\Big{\\{}X_{12,N}\Big{\\}}\eta_{MM^{\prime}}+\text{cyclic
perm. of }(M^{\prime}c^{\prime},Mb,Nc)\right)\ .$ (4.24) At the pole,
$\displaystyle\gamma_{78}=\frac{d}{2}-m,\quad\gamma_{12}=\frac{d}{2}-n-n_{12},\quad\gamma_{34}=\frac{d}{2}-m^{\prime}-n_{34},\quad\gamma_{56}=\frac{d}{2}-n^{\prime\prime}-n_{56}\
,$ (4.25a) with
$\displaystyle\Delta_{7}+\Delta_{8}+1-2\gamma_{78}=\sum_{i=1}^{6}\Delta_{i}+3-2\sum_{i<j}^{6}\gamma_{ij}\
,$ (4.25b) we have
$\displaystyle\sum_{\sum_{i<j}^{6}n_{ij}=m}\prod_{i<j}^{6}\frac{(\gamma_{ij})_{n_{ij}}}{n_{ij}!}$
(4.25c) $\displaystyle=$
$\displaystyle\sum_{n_{12}=0}^{m}\sum_{n_{34}=0}^{m-n_{12}}\sum_{n_{56}=0}^{m-n_{12}-n_{34}}\frac{(\frac{d}{2}-n-n_{12})_{n_{12}}(\frac{d}{2}-m^{\prime}-n_{34})_{n_{34}}(\frac{d}{2}-n^{\prime}-n_{56})_{n_{56}}}{n_{12}!n_{34}!n_{56}!}$
$\displaystyle\times\frac{(d-1-m+n+m^{\prime}+n^{\prime}+n_{12}+n_{34}+n_{56})_{m-n_{12}-n_{34}-n_{56}}}{(m-n_{12}-n_{34}-n_{56})!}\
.$ Shift $n\to n-n_{12}$, $m^{\prime}\to m^{\prime}-n_{34}$ and $n^{\prime}\to
n^{\prime}-n_{56}$
$\left(\sum_{n_{12}=0}^{m}\sum_{n_{34}=0}^{m-n_{12}}\sum_{n_{56}=0}^{m-n_{12}-n_{34}}\frac{(\frac{d}{2}-n-n_{12})_{n_{12}}(\frac{d}{2}-m^{\prime}-n_{34})_{n_{34}}(\frac{d}{2}-n^{\prime}-n_{56})_{n_{56}}}{n_{12}!n_{34}!n_{56}!}\right.\\\
\times\frac{(d-1-m+n+m^{\prime}+n^{\prime}+n_{12}+n_{34}+n_{56})_{m-n_{12}-n_{34}-n_{56}}}{(m-n_{12}-n_{34}-n_{56})!}{\mathbf{{V}}}_{4}^{m^{\prime},n,n^{\prime},0}\frac{\left(\frac{d}{2}-n^{\prime}\right)_{n^{\prime}}}{n^{\prime}!\Gamma\left(\frac{d}{2}+n^{\prime}\right)(\gamma_{56}+n_{56}-\frac{d}{2}+n^{\prime})}\\\
\times\left.\frac{\left(\frac{d}{2}-m^{\prime}\right)_{m^{\prime}}}{m^{\prime}!\Gamma\left(\frac{d}{2}+m^{\prime}\right)(\gamma_{34}+n_{34}-\frac{d}{2}+m^{\prime})}\frac{\left(\frac{d}{2}-n\right)_{n}}{n!\Gamma\left(\frac{d}{2}+n\right)(\gamma_{12}+n_{12}-\frac{d}{2}+n)}\right)\\\
\to\left({\mathbf{{V}}}_{4}^{m^{\prime},n,n^{\prime},m}\frac{\left(\frac{d}{2}-n^{\prime}\right)_{n^{\prime}}}{n^{\prime}!\Gamma\left(\frac{d}{2}+n^{\prime}\right)(\gamma_{56}-\frac{d}{2}+n^{\prime})}\frac{\left(\frac{d}{2}-m^{\prime}\right)_{m^{\prime}}}{m^{\prime}!\Gamma\left(\frac{d}{2}+m^{\prime}\right)(\gamma_{34}-\frac{d}{2}+m^{\prime})}\frac{\left(\frac{d}{2}-n\right)_{n}}{n!\Gamma\left(\frac{d}{2}+n\right)(\gamma_{12}-\frac{d}{2}+n)}\right)\
,$ (4.25d) where ${\mathbf{{V}}}_{4}^{m^{\prime},n,n^{\prime},m}\\\ \equiv
m!\sum_{n_{12}=0}^{\min\\{m,n\\}}\sum_{n_{34}=0}^{\min\\{m-n_{34},m^{\prime}\\}}\sum_{n_{56}=0}^{\min\\{m-n_{12}-n_{34},n^{\prime}\\}}\frac{(d-1-m+n+m^{\prime}+n^{\prime})_{m-n_{12}-n_{34}-n_{56}}}{n_{12}!n_{34}!n_{56}!(m-n_{12}-n_{34}-n_{56})!}\\\
\times{\mathbf{{V}}}_{4}^{m^{\prime}-n_{34},n-n_{12},n^{\prime}-n_{56},0}(n^{\prime}-n_{56}+1)_{n_{56}}\left(\frac{d}{2}+n^{\prime}-n_{56}\right)_{n_{56}}\\\
\times(m^{\prime}-n_{34}+1)_{n_{34}}\left(\frac{d}{2}+m^{\prime}-n_{34}\right)_{n_{34}}(n-n_{12}+1)_{n_{12}}\left(\frac{d}{2}+n-n_{12}\right)_{n_{12}}\
.$ (4.25e)

One can check that from this definition
${\mathbf{{V}}}_{4}^{m^{\prime},n,m,0}={\mathbf{{V}}}_{4}^{m^{\prime},n,0,m}={\mathbf{{V}}}_{4}^{m^{\prime},0,n,m}={\mathbf{{V}}}_{4}^{0,m^{\prime},n,m}$
and reduces to ${\mathbf{{V}}}_{4}^{m^{\prime},n,m,0}$ in (4.21d). With such a
definition of ${\mathbf{{V}}}_{4}^{m^{\prime},n,n^{\prime},m}$, we can rewrite
the Mellin amplitude (4.24) as

$\mathscr{M}_{\texttt{Drone}}^{M_{1}M_{2}\cdots
M_{8}}=-ig^{6}\frac{\pi^{h}}{2}\left(\prod_{i=1}^{8}\mathfrak{D}^{M_{i}A_{i}}\right)\sum_{n^{\prime},n,m^{\prime},m=0}^{\infty}{\mathbf{{V}}}_{4}^{m^{\prime},n,n^{\prime},m}\frac{\Big{\\{}X_{78}^{M}\Big{\\}}{\mathbf{{V}}}_{3}^{m,0,0}}{4m!\Gamma\left(\frac{d}{2}+m\right)(\gamma_{78}-\frac{d}{2}+m)}\\\
\times\frac{\Big{\\{}X_{56}^{M^{\prime}}\Big{\\}}{\mathbf{{V}}}_{3}^{n^{\prime},0,0}}{4n^{\prime}!\Gamma\left(\frac{d}{2}+n^{\prime}\right)(\gamma_{56}-\frac{d}{2}+n^{\prime})}\times\frac{\Big{\\{}X_{34}^{N}\Big{\\}}{\mathbf{{V}}}_{3}^{m^{\prime},0,0}}{4m^{\prime}!\Gamma\left(\frac{d}{2}+m^{\prime}\right)(\gamma_{34}-\frac{d}{2}+m^{\prime})}\\\
\times\frac{\Big{\\{}X_{12}^{M^{\prime\prime}}\Big{\\}}{\mathbf{{V}}}_{3}^{n,0,0}}{4n!\Gamma\left(\frac{d}{2}+n\right)(\gamma_{12}-\frac{d}{2}+n)}f^{a_{7}a_{8}b}f^{a_{5}a_{6}c^{\prime}}f^{a_{3}a_{4}c}f^{a_{1}a_{2}b^{\prime\prime}}\\\
\times\left(\left(f^{c^{\prime}b^{\prime\prime}b^{\prime}}f^{bcb^{\prime}}+f^{bb^{\prime\prime}b^{\prime}}f^{c^{\prime}cb^{\prime}}\right)\eta_{MM^{\prime}}\eta_{NM^{\prime\prime}}+\text{cyclic
perm. of }(M^{\prime}c^{\prime},Mb,Nc)\right)\ ,$ (4.26)

which is related to the flat-space counterpart by the following replacements
up to an overall $\frac{\pi^{h}}{2}\prod_{i=1}^{8}\mathfrak{D}^{M_{i}A_{i}}$.
For the momenta, $ik_{i}\to 2P_{i}$. For propagators,

$\frac{i}{(k_{i}+k_{j})^{2}}\to\frac{1}{4m!\Gamma\left(\frac{d}{2}+m\right)\left(\gamma_{ij}-\frac{d}{2}+m\right)}\
,$ (4.27)

with $m$ to be summed over. For the three-vertex coupling constant, $g\to g\
{\mathbf{{V}}}_{3}^{m,0,0}$, and for the four-vertex coupling constant
$g^{2}\to g^{2}\ {\mathbf{{V}}}_{4}^{m^{\prime},n,n^{\prime},m}$.

###  4.6 Dictionary between gluon Mellin amplitude and flat-space gluon
amplitude

Reflecting on the diverse range of examples that we have provided, spanning
from three to eight-point amplitudes, an intriguing similarity emerges between
the Mellin amplitude in Anti-de Sitter (AdS) spaces and the flat-space
amplitude perturbatively derived from the Feynman rules. This correspondence
is not just supeficial and there is a precise dictionary between them, as
shown in Table 1.

The emergence of kinematic variables in scattering amplitudes is traced back
to the derivative terms in the action. Specifically, in flat space, applying
the derivative $\partial_{x^{A}}$ to the Fourier basis $e^{ik_{i}x}$
introduces a factor, $ik^{A}$. In contrast, within AdS space, the operation of
$\partial_{X^{A}}$ on the boundary-to-bulk propagator, represented as
$\mathcal{E}_{\Delta_{i}}^{M_{i}A_{i}}(P_{i},X)$, produces a factor of
$2P_{i}^{A}$. This analogy provides a rationale for the presence of $ik$ and
$2P_{i}$ on respective sides of the established dictionary.

Indeed, the map $2P_{i}\leftrightarrow ik_{i}$ has been substantiated through
several examples, which demonstrate a notable parallelism in their behaviors.
Firstly, both Mellin amplitudes and flat space amplitudes conform to the null
condition, expressed as $P^{2}=0$ and $k^{2}=0$. Additionally, analogues of
momentum conservation, referenced by equations (2.11) and (2.12), are observed
for these boundary points.

For the internal propagator in the flat-space amplitude, i.e.
$i/(\sum_{i}k_{i})^{2}=i/2\sum_{i<j}k_{i}\cdot k_{j}$, we have
$1/\tilde{\sum}_{i<j}\gamma_{ij}$ on the Mellin side, where we have defined

$\tilde{\sum_{i<j}}\gamma_{ij}\equiv
4m!~{}~{}\Gamma\left(\frac{d}{2}+m\right)\left[\sum_{i<j}\gamma_{ij}+\frac{(d-1)}{2}-\frac{1}{2}\sum_{i}(d-1-\delta_{i})+m\right],$
(4.28)

where again $\delta_{i}$ denotes the scaling dimension of $P_{i}$ in the
Mellin amplitude. The map between the Mandelstam invariants, $k_{i}\cdot
k_{j}$ and $\gamma_{ij}$ resembles the relationship in the scalar scenario, as
referenced in Paulos:2011ie .

From (4.28), it is clear that each bulk-to-bulk propagator is associated with
an integer $m$ to be summed over from 0 to $\infty$. Then, for a three-vertex
connecting the propagators associated with integers $m_{1},m_{2}$ and $m_{3}$,
the three-vertex coupling is

$\displaystyle g\leftrightarrow g\ {\mathbf{{V}}}_{3}^{m_{1},m_{2},m_{3}}\ .$
(4.29)

This may contain boundary-to-bulk propagators with $m_{i}=0$. Similarly, for a
four-vertex connecting propagators associated with integers
$m_{1},m_{2},m_{3}$ and $m_{4}$, the coupling is supposed to be

$\displaystyle g^{2}\leftrightarrow g^{2}\
{\mathbf{{V}}}_{4}^{m_{1},m_{2},m_{3},m_{4}}\ .$ (4.30)

It is also noteworthy that in scalar cases a high energy limit
$\gamma_{ij}\to\infty$ takes Mellin amplitudes to flat-space amplitudes up to
a transform Fitzpatrick:2011ia . Specifically, as reviewed in Appendix C.1, in
the flat-space limit an $n$-point scalar Mellin amplitude becomes

$\begin{split}\mathscr{M}_{n}(\Delta_{i},\gamma_{ij})\approx\frac{\pi^{h}}{2}\prod_{i=1}^{n}\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\int_{0}^{\infty}d\beta\beta^{\frac{1}{2}(\sum_{i=1}^{n}\Delta_{i}-d)-1}e^{-\beta}\mathscr{A}_{n}(p_{i}\cdot
p_{j}=2\beta\gamma_{ij})\ .\end{split}$ (4.31)

It is natural to generalize the flat-space limit to gluon Mellin amplitudes.
Following the dictionary Table 1, we propose that the generalization of flat-
space limit is

$\begin{split}\mathscr{M}_{n}^{M_{1}\cdots
M_{n}}\approx&\frac{\pi^{h}}{2}\left(\prod_{i=1}^{n}\mathfrak{D}^{M_{i}A_{i}}\right)\int_{0}^{\infty}d\beta\beta^{\frac{\sum_{i=1}^{n}\Delta_{i}-d}{2}-1}e^{-\beta}\\\
&\times\mathscr{A}_{n,A_{1}A_{2}\cdots
A_{n}}\bigg{(}\frac{i}{\big{(}\sum_{i}k_{i}\big{)}^{2}}\to\frac{1}{4\beta\sum_{i<j}\gamma_{ij}},ik_{i}\to
2\sqrt{\beta}P_{i}\bigg{)}\ .\end{split}$ (4.32)

We perform several checks for this formula in Appendix C.2.

##  5 Conclusion and outlook

In this study, we present a rigorous computation of the gluon amplitude in
Anti-de Sitter (AdS) space. We employed the embedding formalism, Mellin space
techniques, and an approach utilizing differential operators, successfully
computing novel higher-point correlators. Despite the complexity of the
intermediate steps, we distilled the Mellin results into remarkably succinct
expressions. Intriguingly, we observed that our results exhibit a striking
resemblance to structures in flat space. This work potentially opens new
avenues for research.

It is conceivable that one could rewrite these expressions in spinor helicity
formalism. Conventionally, flat space scattering amplitudes are written using
polarization vectors. However, the preceding twenty years have witnessed
advancements through the adoption of spinor helicity variables. These
variables are distinguished by their transformation properties under the
spinor representations of both the Lorentz group and the little group. It
would be interesting to see if one could use a different variable to simplify
the expressions.777For a bispinor formalism of AdS correlators in embedding
space, see Binder:2020raz

In the context of Mellin space, the domain of external gravitons presents a
significant avenue for further exploration. Only a handful of works have been
conducted in this direction Bonifacio:2022vwa ; Raju:2012zs ; Albayrak:2019yve
; Albayrak:2023jzl . The four-point external graviton falls within our
investigative purview. Moreover, the potential to establish a map between flat
space graviton scattering and graviton bulk scattering in the AdS framework is
both promising and of considerable practical relevance from AdS as well as dS
point of view.

Related to the graviton Mellin amplitudes, several research groups have made
progress in the (A)dS color kinematics and double copy frontier, though
predominantly limited to three and four-point configurations Armstrong:2020woi
; Albayrak:2020fyp ; Alday:2021odx ; Diwakar:2021juk ;
Sivaramakrishnan:2021srm ; Cheung:2022pdk ; Herderschee:2022ntr ;
Farrow:2018yni ; Lipstein:2019mpu ; Jain:2021qcl ; Zhou:2021gnu ; Lee:2022fgr
; Bissi:2022wuh ; Armstrong:2023phb ; Alday:2022lkk ; Liang:2023zxo . To truly
harness the potential for double copy, it is important to systematically
explore higher-point configurations! Digging deeper into these intricate
structures not only broadens the range of computable amplitudes but also
underscores the efficacy and profound insights offered by the double copy
approach. Our method for tackling higher-point configurations and its
resemblance to flat space could be important for constructing the double copy
that mirrors the flat-space version.

An exciting opportunity presents itself in exploring the computation of
spinning loops within AdS (see some work in this direction Giombi:2017hpr ;
Albayrak:2020bso ; Huang:2023ppy ). Previous investigations into loop
calculations in flat space have revealed crucial links between trees and
loops, bridging gravitational theories with gauge theories. In flat space,
these loop amplitudes also exhibit connections to intricate geometric
structures. It would be fascinating to investigate whether similar patterns or
connections emerge in the loops within AdS.

## Acknowledgement

The authors express sincere thanks for the enlightening discussions with Soner
Albayrak. We thank Xinkang Wang for comment on the draft. We thank developers
of Tikz, which was used in drawing the diagrams in this paper.

##  Appendix A Three-point scalar and gluon amplitudes: Schwinger trick

In this appendix, we review the calculation of the three-point AdS amplitudes
Paulos:2011ie . For the scalar case, as discussed above, the correlation
function can be calculated by integrating the bulk-to-boundary propagators,
i.e.

$\displaystyle\left\langle\mathcal{O}_{1}(P_{1})\mathcal{O}_{2}(P_{2})\mathcal{O}_{3}(P_{3})\right\rangle=ig\int_{\text{AdS}}dX\
\mathcal{E}(P_{1},X)~{}\mathcal{E}(P_{2},X)\mathcal{E}(P_{3},X)\ .$ (A.1)

To perform the integration, it is convenient to express the bulk-to-boundary
propagator with Schwinger parameter,

$\displaystyle\mathcal{E}(P_{i},X)=\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\int_{0}^{+\infty}\frac{dt_{i}}{t_{i}}t_{i}^{\Delta_{i}}e^{2t_{i}P_{i}\cdot
X}\ .$ (A.2)

Then, the integration becomes

$\displaystyle\left\langle\mathcal{O}_{1}(P_{1})\mathcal{O}_{2}(P_{2})\mathcal{O}_{3}(P_{3})\right\rangle=ig\prod_{i=1}^{3}\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\int_{0}^{+\infty}\frac{dt_{i}}{t_{i}}t^{\Delta_{i}}\int_{\text{AdS}}dXe^{2T\cdot
X}\ ,$ (A.3)

where $T\equiv\sum_{i=1}^{3}t_{i}P_{i}$. Since $t_{i}$ are positive and
$P_{i}$ are null vectors, $T$ must be time-like. In the rest frame where
$T=(T^{0},T^{\mu})=(|T|,0)$, parametrize the AdSd+1 space by

$\displaystyle
X=\left(\frac{1+x_{0}^{2}+x^{2}}{2x_{0}},\frac{1-x_{0}^{2}-x^{2}}{2x_{0}},x^{\mu}\right)\
,$ (A.4)

which satisfies the equation (2.1) for $R=1$. Then,

$\displaystyle\int_{\text{AdS}}dXe^{2T\cdot X}=$
$\displaystyle\int_{0}^{+\infty}\frac{dx_{0}}{x_{0}^{d+1}}\int_{0}^{+\infty}d^{d}xe^{-\frac{1+x_{0}^{2}+x^{2}}{x_{0}}|T|}$
(A.5) $\displaystyle=$
$\displaystyle\pi^{h}\int_{0}^{+\infty}\frac{dx_{0}}{x_{0}^{h+1}}e^{-x_{0}+\frac{T^{2}}{x_{0}}}\
.$

So,

$\displaystyle\left\langle\mathcal{O}_{1}(P_{1})\mathcal{O}_{2}(P_{2})\mathcal{O}_{3}(P_{3})\right\rangle=$
$\displaystyle
ig\pi^{h}\prod_{i=1}^{3}\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\int_{0}^{+\infty}\frac{dt_{i}}{t_{i}}t^{\Delta_{i}}\int_{0}^{+\infty}\frac{dx_{0}}{x_{0}^{h+1}}e^{-x_{0}+\frac{T^{2}}{x_{0}}}$
(A.6) $\displaystyle=$ $\displaystyle
ig\pi^{h}\prod_{i=1}^{3}\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\int_{0}^{+\infty}\frac{dt_{i}}{t_{i}}t^{\Delta_{i}}e^{T^{2}}\int_{0}^{+\infty}\frac{dx_{0}}{x_{0}^{h-\frac{\sum_{i=1}^{3}\Delta_{i}}{2}+1}}e^{-x_{0}}$
$\displaystyle=$ $\displaystyle
ig\pi^{h}\Gamma\left(\frac{\sum_{i=1}^{3}\Delta_{i}-d}{2}\right)\prod_{i=1}^{3}\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\int_{0}^{+\infty}\frac{dt_{i}}{t_{i}}t_{i}^{\Delta_{i}}e^{-t_{1}t_{2}P_{12}-t_{1}t_{3}P_{13}-t_{2}t_{3}P_{23}}\
,$

where $P_{ij}\equiv-2P_{i}\cdot P_{j}$.

For the integration over Schwinger parameters $t_{i}$, we change the variables
$m_{ij}=t_{i}t_{j}$. Then,

$\displaystyle\left\langle\mathcal{O}_{1}(P_{1})\mathcal{O}_{2}(P_{2})\mathcal{O}_{3}(P_{3})\right\rangle=$
$\displaystyle
ig\frac{\pi^{h}}{2}\Gamma\left(\frac{\sum_{i=1}^{3}\Delta_{i}-d}{2}\right)\prod_{i=1}^{3}\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\prod_{i<j}^{3}\int_{0}^{+\infty}\frac{dm_{ij}}{m_{ij}}m_{ij}^{\gamma_{ij}}e^{-m_{ij}P_{ij}}$
(A.7) $\displaystyle=$ $\displaystyle
ig\frac{\pi^{h}}{2}\Gamma\left(\frac{\sum_{i=1}^{3}\Delta_{i}-d}{2}\right)\prod_{i=1}^{3}\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\prod_{i<j}^{3}\Gamma(\gamma_{ij})P_{ij}^{-\gamma_{ij}}\
,$

which gives the three-point scalar Mellin amplitude (2.6).

Similarly, for the three-point gluon amplitude, we have

$\left\langle J_{1}(P_{1})J_{2}(P_{2})J_{3}(P_{3})\right\rangle\\\
=-igf^{a_{1}a_{2}a_{3}}\int_{\text{AdS}}dX\bigg{[}\eta_{A_{1}A_{2}}\left(\frac{\partial}{\partial
X_{A_{3}}}\mathcal{E}^{M_{2}A_{2}}(P_{2},X)\right)\mathcal{E}^{M_{2}A_{2}}(P_{2},X)\mathcal{E}^{M_{3}A_{3}}(P_{3},X)\\\
-(1\leftrightarrow 2)+\text{cyclic permutations}\bigg{]}\\\
=-igf^{a_{1}a_{2}a_{3}}\prod_{i=1}^{3}\mathfrak{D}^{M_{i}A_{i}}\Big{[}2\eta_{A_{1}A_{2}}P_{2,A_{3}}\prod_{i=1}^{3}\int_{0}^{+\infty}\frac{dt_{i}}{t_{i}}t^{\Delta_{1}}t^{\Delta_{2}+1}t^{\Delta_{3}}\int_{\text{AdS}}dXe^{2T\cdot
X}\\\ -(1\leftrightarrow 2)+\text{cyclic permutations}\Big{]}\\\
=\frac{\pi^{h}}{2}\Gamma\left(\frac{\sum_{i=1}^{3}\Delta_{i}-d+1}{2}\right)igf^{a_{1}a_{2}a_{3}}\prod_{i=1}^{3}\mathfrak{D}^{M_{i}A_{i}}\mathscr{I}_{A_{1}A_{2}A_{3}}\prod_{i<j}^{3}\Gamma(\gamma_{ij})P_{ij}^{-\gamma_{ij}}\
,$ (A.8)

from which the Mellin amplitude, (3.2), can be read off.

##  Appendix B Symanzik’s Formula

As seen in this paper, the Symanzik’s formula is crucial for calculating
higher point amplitude from lower point amplitudes. Here we review the
derivation of this formula.

First, with Schwinger parameters, the l.h.s. of (3.10) can be written as

$\displaystyle\int_{\partial\text{AdS}}dQ\prod_{i=1}^{n}\Gamma(\ell_{i})\left(-2P_{i}\cdot
Q\right)^{-\ell_{i}}=\prod_{i=1}^{n}\int_{0}^{+\infty}\frac{dt_{i}}{t_{i}}t^{\ell_{i}}\int_{\partial\text{AdS}}dQe^{2T\cdot
Q}\ ,$ (B.1)

where $T\equiv\sum_{i=1}^{n}t_{i}P_{i}$. Since $t_{i}$ are positive and
$P_{i}$ are null vectors, $T$ must be time-like. In the rest frame where
$T=(T^{0},T^{\mu})=(|T|,0)$, parametrize the boundary of AdSd+1 by

$\displaystyle Q=\left(\frac{x^{2}+1}{2},\frac{x^{2}-1}{2},x^{\mu}\right)\ .$
(B.2)

Then,

$\displaystyle\int_{\partial\text{AdS}}dQe^{2T\cdot
Q}=\frac{\pi^{h}}{|T|^{h}}e^{-|T|}\ ,$ (B.3)

where $|T|=\sqrt{-\sum_{i,j=1}^{n}t_{i}t_{j}P_{i}\cdot P_{j}}$.

Change variables $t_{i}\to t_{i}|T|$. Then,

$\displaystyle\prod_{i=1}^{n}\int_{0}^{+\infty}\frac{dt_{i}}{t_{i}}t^{\ell_{i}}\int_{\partial\text{AdS}}dQe^{2T\cdot
Q}=\pi^{h}\int_{0}^{+\infty}\frac{\det(\partial_{t_{i}}t_{j}|T|)}{|T|^{n}}\frac{|T|^{\sum_{i=1}^{n}\ell_{i}}}{|T|^{d}}\prod_{i=1}^{n}\frac{dt_{i}}{t_{i}}t^{\ell_{i}}e^{-|T|^{2}}\
.$ (B.4)

The Jacobian is

$\displaystyle\det(\partial_{t_{i}}t_{j}|T|)=\det(\delta_{ij}|T|-t_{j}\frac{2\sum_{k=1}^{n}t_{k}P_{k}\cdot
P_{i}}{2|T|})=|T|^{n}\left(1-\frac{\sum_{k,i=1}^{n}t_{k}t_{i}P_{k}\cdot
P_{i}}{|T|^{2}}\right)=2|T|^{n}\ ,$ (B.5)

where in the second step we have used the formula
$\det(\delta_{ij}+A_{i}B_{j})=1+\sum_{i}A_{i}B_{i}$. For
$\sum_{i=1}^{n}\ell_{i}=d$, (B.4) becomes

$\displaystyle
2\pi^{h}\int_{0}^{+\infty}\prod_{i=1}^{n}\frac{dt_{i}}{t_{i}}t^{\ell_{i}}e^{-|T|^{2}}=2\pi^{h}\int_{0}^{+\infty}\prod_{i=1}^{n}\frac{dt_{i}}{t_{i}}t^{\ell_{i}}e^{-\sum_{i<j}^{n}t_{i}t_{j}P_{ij}}\
.$ (B.6)

Recall that in (A.7) we have derived that

$\displaystyle
2\pi^{h}\prod_{i=1}^{3}\int_{0}^{+\infty}\frac{dt_{i}}{t_{i}}t_{i}^{\Delta_{i}}e^{-t_{1}t_{2}P_{12}-t_{1}t_{3}P_{13}-t_{2}t_{3}P_{23}}=\pi^{h}\prod_{i<j}^{3}\Gamma(\gamma_{ij})P_{ij}^{-\gamma_{ij}}\
.$ (B.7)

More generally,

$\displaystyle
2\pi^{h}\prod_{i=1}^{n}\int_{0}^{+\infty}\frac{dt_{i}}{t_{i}}t_{i}^{\ell_{i}}e^{-\sum_{i<j}^{n}t_{i}t_{j}P_{ij}}=\pi^{h}\int\prod_{i<j}^{n}\frac{d\gamma_{ij}}{2\pi
i}\Gamma(\gamma_{ij})P_{ij}^{-\gamma_{ij}}\prod_{i=1}^{n}\delta\left(\sum_{j\neq
i}^{n}\gamma_{ij}-\ell_{i}\right)\ ,$ (B.8)

and thus proves the Symanzik’s formula (3.10).

##  Appendix C Flat-space limit

###  C.1 Review: flat-space limit of scalar correlators

In Fitzpatrick:2011ia , it was shown that in the limit of
$\gamma_{ij}\to\infty$, a scalar Mellin amplitude
$\mathscr{M}(\Delta_{i},\gamma_{ij})$ reproduces the flat-space amplitude
$\mathscr{A}(p_{i})$. In particular, in the case of massless scalar
scattering, the flat-space limit is given by

$\begin{split}\mathscr{M}_{n}(\Delta_{i},\gamma_{ij})\approx\frac{\pi^{h}}{2}\prod_{i=1}^{n}\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\int_{0}^{\infty}d\beta\beta^{\frac{1}{2}(\sum_{i=1}^{n}\Delta_{i}-d)-1}e^{-\beta}\mathscr{A}_{n}(p_{i}\cdot
p_{j}=2\beta\gamma_{ij})\ .\end{split}$ (C.1)

For example, in $\phi^{3}$ theory, this relation is simply true for three-
point amplitude without even taking the limit (actually the Mellin variables
are fixed for this case, so no limit can be taken). To see that, we plug
$\mathscr{A}_{3}=g$ in the r.h.s. of (C.1) and get

$\begin{split}g\frac{\pi^{h}}{2}\prod_{i=1}^{3}\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\int_{0}^{\infty}d\beta\beta^{\frac{1}{2}(\sum_{i=1}^{3}\Delta_{i}-d)-1}e^{-\beta}=g\frac{\pi^{\frac{d}{2}}}{2}\prod_{i=1}^{3}\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\Gamma\left(\frac{\sum_{i=1}^{3}\Delta_{i}-d}{2}\right)\
,\end{split}$ (C.2)

which is precisely the scalar three-point Mellin amplitude (2.6). A slightly
nontrivial example is the four-point amplitude. In the $s$-channel with the
exchange field of scaling dimension $\Delta$, the Mellin amplitude is
Paulos:2011ie

$\mathscr{M}_{\texttt{Exch}}(\Delta_{i})=\frac{\pi^{h}}{2}\prod_{i=1}^{4}\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\sum_{n=0}^{\infty}\frac{g^{2}}{4n!\
\Gamma(1+\Delta-h+n)\big{(}\gamma_{12}-\frac{\Delta_{1}+\Delta_{2}-\Delta}{2}+n\big{)}}\\\
\times\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+\Delta-d}{2}\right)\left(1-\frac{\Delta_{1}+\Delta_{2}-\Delta}{2}\right)_{n}\\\
\times\Gamma\left(\frac{\Delta_{3}+\Delta_{4}+\Delta-d}{2}\right)\left(1-\frac{\Delta_{3}+\Delta_{4}-\Delta}{2}\right)_{n}\
.$ (C.3)

In the flat-space limit, $\gamma_{12}\to\infty$,

$\mathscr{M}_{\texttt{Exch}}(\Delta_{i})\approx\frac{\pi^{h}}{2}\prod_{i=1}^{4}\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\frac{g^{2}}{4\gamma_{12}}\sum_{n=0}^{\infty}\frac{1}{n!\Gamma(1+\Delta-d/2+n)}\\\
\times\left.\left(\frac{\partial^{n}}{\partial
t_{1}^{n}}\int_{0}^{\infty}d\beta_{1}\beta_{1}^{\frac{1}{2}(\Delta_{1}+\Delta_{2}+\Delta-d)-1}e^{-\beta_{1}}t_{1}^{\frac{1}{2}(\Delta_{1}+\Delta_{2}-\Delta)-1}\right)\right|_{t_{1}=1}\\\
\times\left.\left(\frac{\partial^{n}}{\partial
t_{2}^{n}}\int_{0}^{\infty}d\beta_{2}\beta_{2}^{\frac{1}{2}(\Delta_{3}+\Delta_{4}+\Delta-d)-1}e^{-\beta_{2}}t_{2}^{\frac{1}{2}(\Delta_{3}+\Delta_{4}-\Delta)-1}\right)\right|_{t_{2}=1}\
.$ (C.4)

Rescaling the integration variables $\beta_{1,2}\to\beta_{1,2}/t_{1,2}$ and
using the following identity proven in Appendix C.3 of Fitzpatrick:2011ia ,
namely

$\sum_{n=0}^{\infty}\frac{1}{n!\Gamma(1+\Delta-d/2+n)}\left.\left(\frac{\partial^{n}}{\partial
t_{1}^{n}}\frac{\partial^{n}}{\partial
t_{2}^{n}}e^{-\frac{\beta_{1}}{t_{1}}-\frac{\beta_{2}}{t_{2}}}t_{1}^{-\Delta+\frac{d}{2}-1}t_{2}^{-\Delta+\frac{d}{2}-1}\right)\right|_{t_{1}=t_{2}=1}=\\\
\beta_{1}^{\frac{d}{2}-\Delta}e^{-\beta_{1}}\delta(\beta_{1}-\beta_{2})\ ,$
(C.5)

we get

$\begin{split}\mathscr{M}_{\texttt{Exch}}(\Delta_{i})\approx&\frac{\pi^{h}}{2}\prod_{i=1}^{4}\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})}\int_{0}^{\infty}d\beta\beta^{\frac{1}{2}(\sum_{i=1}^{4}\Delta_{i}-d)-1}\frac{g^{2}}{4\beta\gamma_{12}}e^{-\beta}\
.\end{split}$ (C.6)

Note that it is consistent with (C.1) with
$\mathscr{A}_{\texttt{Exch}}(p_{i})=g^{2}(2p_{1}\cdot p_{2})^{-1}$ the flat-
space amplitude in the $s$-channel.

###  C.2 Spinning flat space limits

Now we try to generalize (C.1) to the vector case. In the previous sections we
have already seen the similarity between the gluon amplitude and the flat-
space counterpart. In particular, we conjecture the dictionaries between them.
So, it is natural to expect that in the flat-space limit Mellin amplitudes
reduce to the flat-space amplitudes.

First, for the three-point gluon amplitude (3.2), we have the relation

$\mathscr{M}_{\texttt{3v}}^{M_{1}M_{2}M_{3}}=\frac{\pi^{h}}{2}\left(\prod_{i=1}^{3}\mathfrak{D}^{M_{i}A_{i}}\right)\int_{0}^{\infty}d\beta\beta^{\frac{1}{2}(\gamma_{1}+\gamma_{2}+\gamma_{3}-d)-1}e^{-\beta}\mathscr{A}_{3,A_{1}A_{2}A_{3}}(ik_{i}\to
2\sqrt{\beta}P_{i})\ .$ (C.7)

For the four-point gluon amplitude of the contact diagram, the Mellin
amplitude (3.18b) can also be expressed as

$\begin{split}\mathscr{M}^{M_{1}M_{2}M_{3}M_{4}}_{\texttt{Exch}}=\frac{\pi^{h}}{2}\left(\prod_{i=1}^{4}\mathfrak{D}^{M_{i}A_{i}}\right)\int_{0}^{\infty}d\beta\beta^{\frac{1}{2}(\sum_{i=0}^{4}\Delta_{i}-d)-1}e^{-\beta}\mathscr{A}_{\text{contact},A_{1}A_{2}A_{3}A_{4}}\
.\end{split}$ (C.8)

Now we look at the $s$-channel of the vector four-point Mellin amplitude
(3.13). In the limit of $\gamma_{12}\to\infty$, it becomes

$\mathscr{M}^{M_{1}M_{2}M_{3}M_{4}}_{\texttt{Exch}}\approx-g^{2}\frac{\pi^{h}}{2}f^{a_{1}a_{2}b}f^{a_{3}a_{4}b}\left(\prod_{i=1}^{4}\mathfrak{D}^{M_{i}A_{i}}\right)\sum_{n=0}^{\infty}\frac{\left(\Gamma\left(d-1\right)(\frac{d}{2}-n)_{n}\right)^{2}}{4n!\Gamma\left(\frac{d}{2}+n\right)\gamma_{12}}\Big{\\{}X_{12}\Big{\\}}\cdot\Big{\\{}X_{34}\Big{\\}}\
.$ (C.9)

The sum over $n$ can be implemented by using (C.5) with $\Delta=d-1$. So,

$\sum_{n=0}^{\infty}\frac{\left[\Gamma\left(d-1\right)(\frac{d}{2}-n)_{n}\right]^{2}}{n!\Gamma\left(\frac{d}{2}+n\right)}=\sum_{n=0}^{\infty}\frac{1}{n!\Gamma(d/2+n)}\left.\left(\frac{\partial^{n}}{\partial
t_{1}^{n}}\int_{0}^{\infty}d\beta_{1}\beta_{1}^{d-2}e^{-\beta_{1}}t_{1}^{\frac{d}{2}-1}\right)\right|_{t_{1}=1}\\\
\times\left.\left(\frac{\partial^{n}}{\partial
t_{2}^{n}}\int_{0}^{\infty}d\beta_{2}\beta_{2}^{d-2}e^{-\beta_{2}}t_{2}^{\frac{d}{2}-1}\right)\right|_{t_{2}=1}=\int_{0}^{\infty}d\beta\beta^{\frac{3d-4}{2}-1}e^{-\beta}\
.$ (C.10)

Therefore,

$\begin{split}\mathscr{M}^{M_{1}M_{2}M_{3}M_{4}}_{\texttt{Exch}}\approx&-g^{2}\frac{\pi^{h}}{2}f^{a_{1}a_{2}b}f^{a_{3}a_{4}b}\left(\prod_{i=1}^{4}\mathfrak{D}^{M_{i}A_{i}}\right)\int_{0}^{\infty}d\beta\beta^{\frac{3d-4}{2}-1}e^{-\beta}\frac{\mathscr{I}_{A_{1}A_{2}A_{3}A_{4}}}{4\gamma_{12}}\\\
=&\frac{\pi^{h}}{2}\left(\prod_{i=1}^{4}\mathfrak{D}^{M_{i}A_{i}}\right)\int_{0}^{\infty}d\beta\beta^{\frac{3d-4}{2}-1}e^{-\beta}\\\
&\times\mathscr{A}_{s-\text{channel},A_{1}A_{2}A_{3}A_{4}}\left(\frac{i}{k_{1}\cdot
k_{2}}\to\frac{1}{2\beta\gamma_{12}},ik_{i}\to 2\sqrt{\beta}P_{i}\right)\
.\end{split}$ (C.11)

Combining (C.7), (C.8) and (C.11), we can summarize the flat-space limit for
gluon Mellin amplitudes as follows. Namely, as $\gamma_{ij}\to\infty$,

$\begin{split}\mathscr{M}_{n}^{M_{1}\cdots
M_{n}}\approx&\frac{\pi^{h}}{2}\left(\prod_{i=1}^{n}\mathfrak{D}^{M_{i}A_{i}}\right)\int_{0}^{\infty}d\beta\beta^{\frac{\sum_{i=1}^{n}\Delta_{i}-d}{2}-1}e^{-\beta}\\\
&\times\mathscr{A}_{n,A_{1}A_{2}\cdots
A_{n}}\bigg{(}\frac{i}{\big{(}\sum_{i}k_{i}\big{)}^{2}}\to\frac{1}{4\beta\sum_{i<j}\gamma_{ij}},ik_{i}\to
2\sqrt{\beta}P_{i}\bigg{)}\ .\end{split}$ (C.12)

Now we show a partial proof of (C.12) for the case where the $(n+1)$-point
amplitude can be factorized into an $n$-point amplitude and a three-point
amplitude, as in (4.1) (also see Figure 4). First, if the $n$-point amplitude
satisfies (C.12) in the flat-space limit, we can write

$\tilde{\mathscr{M}}_{n,A_{1}A_{2}\cdots
A_{n}}(P_{1},P_{2},\cdots,P_{n-1},\gamma_{ij})\approx\int_{0}^{\infty}d\beta\beta^{\frac{(n-1)d-n}{2}-1}e^{-\beta}\\\
\times\mathscr{A}_{n,A_{1}A_{2}\cdots
A_{n}}\bigg{(}\frac{i}{\big{(}\sum_{i}k_{i}\big{)}^{2}}\to\frac{1}{4\beta\sum_{i<j}\gamma_{ij}},ik_{i}\to
2\sqrt{\beta}P_{i}\bigg{)}\ .$ (C.13)

By plugging it in (4.7), we get

$\mathscr{M}_{n+1}^{M_{1}M_{2}\cdots
M_{n+1}}\approx\frac{\pi^{h}}{2}\left(\prod_{i=1}^{n+1}\mathfrak{D}^{M_{i}A_{i}}\right)igf^{a_{n}a_{n+1}b}\Big{\\{}X_{n(n+1)}^{M}\Big{\\}}\sum_{m=0}^{\infty}\frac{{\mathbf{{V}}}_{3}^{m,0,0}}{4\Gamma\left(\frac{d}{2}+m\right)\gamma_{n(n+1)}}\\\
\times\sum_{\sum_{r<s}^{n-1}n_{rs}=m}\prod_{r<s}^{n-1}\frac{(\gamma_{rs})_{n_{rs}}}{n_{rs}!}\int_{0}^{\infty}d\beta\beta^{\frac{(n-1)d-n}{2}-1}e^{-\beta}\\\
\times\mathscr{A}_{A_{1}A_{2}\cdots A_{n-1}M}^{a_{1}a_{2}\cdots
a_{n-1}b}\bigg{(}\frac{i}{\big{(}\sum_{i}k_{i}\big{)}^{2}}\to\frac{1}{4\beta\sum_{i<j}(\gamma_{ij}+n_{ij})},ik_{i}\to
2\sqrt{\beta}P_{i}\bigg{)}\ .$ (C.14)

For the sum over $n_{ij}$, we can repeatedly use

$\displaystyle\sum_{\sum_{i<j}n_{ij}\text{
fixed}}\prod_{i<j}\frac{(\gamma_{ij})_{n_{ij}}}{n_{ij}!}\frac{1}{\sum_{i<j}(\gamma_{ij}+n_{ij})}=$
$\displaystyle\frac{(\sum_{i<j}\gamma_{ij})_{\sum_{i<j}n_{ij}}}{(\sum_{i<j}n_{ij})!}\frac{1}{\sum_{i<j}(\gamma_{ij}+n_{ij})}$
(C.15) $\displaystyle\approx$
$\displaystyle\frac{(\sum_{i<j}\gamma_{ij})_{\sum_{i<j}n_{ij}}}{(\sum_{i<j}n_{ij})!}\frac{1}{\sum_{i<j}(\gamma_{ij}+n_{ij})-1}$
$\displaystyle=$
$\displaystyle\frac{(\sum_{i<j}\gamma_{ij}-1)_{\sum_{i<j}n_{ij}}}{(\sum_{i<j}n_{ij})!}\frac{1}{\sum_{i<j}\gamma_{ij}-1}$
$\displaystyle\approx$
$\displaystyle\frac{(\sum_{i<j}\gamma_{ij}-1)_{\sum_{i<j}n_{ij}}}{(\sum_{i<j}n_{ij})!}\frac{1}{\sum_{i<j}\gamma_{ij}}\
,$

for each propagator term in (C.14). Therefore, with totally number of
propagators $N_{p}$, we have

$\sum_{\sum_{r<s}^{n-1}n_{rs}=m}\prod_{r<s}^{n-1}\frac{(\gamma_{rs})_{n_{rs}}}{n_{rs}!}\prod_{\text{propagators}}\frac{1}{\sum_{i<j}(\gamma_{ij}+n_{ij})}\\\
\approx\frac{\big{(}\sum_{r<s}^{n-1}\gamma_{rs}-N_{p}\big{)}_{m}}{m!}\prod_{\text{propagators}}\frac{1}{\sum_{i<j}\gamma_{ij}}\
.$ (C.16)

Here, the sum of $\gamma_{rs}$ can be evaluated at the pole

$\displaystyle\sum_{i<j}^{n-1}\gamma_{ij}=\frac{(n-2)(d-1)-\sum_{i=1}^{n-1}\delta_{i}}{2}-m\
.$ (C.17)

Thus, (C.14) becomes

$\mathscr{M}^{M_{1}M_{2}\cdots
M_{n+1}}_{n+1}\approx\frac{\pi^{h}}{2}\left(\prod_{i=1}^{n+1}\mathfrak{D}^{M_{i}A_{i}}\right)igf^{a_{n}a_{n+1}b}\Big{\\{}X_{n(n+1)}^{M}\Big{\\}}\frac{1}{4\gamma_{n(n+1)}}\sum_{m=0}^{\infty}\frac{1}{m!\Gamma\left(\frac{d}{2}+m\right)}\\\
\times\left.\left(\frac{\partial^{m}}{\partial
t_{1}^{m}}\int_{0}^{\infty}d\beta_{1}\beta_{1}^{d-2}e^{-\beta_{1}}t_{1}^{\frac{d}{2}-1}\right)\right|_{t_{1}=1}\bigg{(}\frac{\partial^{m}}{\partial
t_{2}^{m}}\int_{0}^{\infty}d\beta_{2}\beta_{2}^{\frac{(n-1)d-n}{2}-1}e^{-\beta_{2}}\\\
\mathscr{A}_{A_{1}A_{2}\cdots A_{n-1}M}^{a_{1}a_{2}\cdots
a_{n-1}b}\Big{(}\frac{i}{\big{(}\sum_{i}k_{i}\big{)}^{2}}\to\frac{1}{4\beta_{2}\sum_{i<j}\gamma_{ij}},ik_{i}\to
2\sqrt{\beta_{2}}P_{i}\Big{)}t_{2}^{\frac{(n-2)(d-1)-\sum_{i=1}^{n-1}\delta_{i}}{2}-N_{p}-1}\bigg{)}\bigg{|}_{t_{2}=1}\
.$ (C.18)

Rescaling $\beta_{1}\to\beta_{1}/t_{1}$, $\beta_{2}\to\beta_{2}/t_{2}$ in
(C.18) and using (C.5) with $\Delta=d-1$, we finally arrive at the formula
(C.12) for the $(n+1)$-point Mellin amplitude $\mathscr{M}_{n+1}$.

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