2401.00038

arXiv:2401.00038v1 [hep-th] 29 Dec 2023
Towards the Feynman rule for n-point gluon Mellin
amplitudes in AdS/CFT
Jinwei Chu, Savan Kharel
Department of Physics, University of Chicago, Chicago, IL 60637, USA
E-mail: jinweichu@uchicago.edu, skharel@uchicago.edu
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 higherpoint 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.
Contents
1 Introduction
2
2 Preliminaries and summary
2.1 Embedding space formalism
2.2 Mellin space
2.3 AdS amplitudes and toolkit
2.4 Summary of the main results
3
3
4
5
7
3 Setting the stage: the three, four, and five-point gluon amplitudes
3.1 Three-point gluon amplitude
3.2 Four-point gluon amplitudes
3.2.1 Contact diagram
3.2.2 Exchange diagram
3.3 Five-point gluon amplitudes
3.3.1 Channel with a three-vertex and a four-vertex
3.3.2 Channel with three three-vertices
8
8
9
9
10
11
12
15
4 Higher-point gluon amplitudes
4.1 Factorization of (n + 1)-point gluon amplitudes
4.2 Six-point amplitude: snowflake channel
4.3 Six-point amplitude: channel with two three-vertices and a four-vertex
4.4 Seven-point amplitude: scarecrow channel
4.5 Eight-point amplitude: drone channel
4.6 Dictionary between gluon Mellin amplitude and flat-space gluon amplitude
18
18
21
23
25
27
29
5 Conclusion and outlook
30
A Three-point scalar and gluon amplitudes: Schwinger trick
31
B Symanzik’s Formula
32
C Flat-space limit
C.1 Review: flat-space limit of scalar correlators
C.2 Spinning flat space limits
33
33
35
–1–
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 [1, 2]. 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. [3–7]). Two primary difficulties emerged in
this field: the complexity of bulk integrals and the intricacies involved in dealing with spinbearing 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 [8–32], position-Mellin space [33–47, 47–53], and more
recently momentum-Mellin space [54–56]. 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 [57].
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 [58]. Furthermore, these advancements unveil fascinating connections between core physics and diverse mathematical fields [59, 60]. 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 [61–64]. 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
–2–
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) [65, 66] aligning
well with the program to construct cosmologically relevant correlators [67–70]. 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 higherpoint 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 [71] 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,
⟨O1 (P1 ) · · · On (Pn )⟩ where Pi denotes the AdS boundary coordinate where the operator
Oi 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 [35].1 This formalism stands as a robust tool for the in-depth explo1
In a seminal work by Dirac [72], it was proposed that the conformal group SO(d + 1, 1) naturally
“lives” in the embedding space Rd+1,1 . Here, it can be understood as the group of linear isometries. This
–3–
ration 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 Rd+1,1 . AdS coordinate vectors X satisfy the following property,
X · X ≡ ηM N X M X N = −R2 .
(2.1)
Throughout the paper we will take R = 1. The boundary of the AdSd+1 space is at
X → ∞, 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 · P ≡ ηM N P M P N = 0.
Therefore, the distance between any two boundary points Pi and Pj is defined by Pij ≡
(Pi − Pj )2 = −2Pi · Pj .
2.2
Mellin space
Another key mathematical apparatus utilized in our study is the Mellin space.2 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.
Q
−γ
The basis of Mellin space is i<j Pij ij , where γij are called Mellin variables. The
P
scaling dimension of this basis for Pi is
j̸=i γij . First, we focus on the scalar cases.
Expanded in Mellin space, an n-point amplitude can be expressed as

* n
+ Z  n
n X
Y
Y dγij
Y
−γij 

Oi (Pi ) =
Γ(γij )Pij
δ
γij − ∆i Mn (γij ) ,
(2.2)
2πi
i=1
i<j
i=1
j̸=i
where Mn (γij ) is called Mellin amplitude. Note that the delta functions restrict the correct
scaling behavior of Oi (Pi ). For the sake of notational simplicity, we will forgo including
them in our subsequent equations.
In the context of vector fields J Mi (Pi ), our primary interest in this paper, the amplitude
takes on a slightly different form to incorporate the indices. We can write it as

+ Z  n
* n
Y dγij
Y
−γ
J Mi (Pi ) = 
Γ(γij )Pij ij  MnM1 M2 ···Mn (γij , Pi ) .
(2.3)
2πi
i=1
i<j
In this context, it is crucial to underline a subtle difference as compared to the scalar
scenario. Specifically, the Mellin amplitude MnM1 M2 ···Mn (γij , Pi ) is a function not only of
the Mellin variables γij , but also of the boundary coordinates Pi . This is attributed to the
possibility that vector Mellin amplitude may contain Pi with free indices.3
suggests that constraints imposed by conformal symmetry could be as straightforward as those from Lorentz
symmetry. Also see Weinberg’s paper [73].
2
See [37] for a nice review of Mellin Space in the AdS/CFT.
P
3
More generally, each field in the correlation function has a spin of ℓi . Then, there are totally n
i=1 ℓi
free indices in the Mellin amplitude.
–4–
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 Oi is a function of the boundary point
Pi and the bulk point X, i.e.,
E(Pi , X) =
C∆i
,
(−2Pi · X)∆i
C∆i =
Γ(∆i )
,
+ 1 − h)
2π h Γ (∆i
(2.4)
where h ≡ 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:
Z
⟨O1 (P1 )O2 (P2 )O3 (P3 )⟩ = ig
dX E(P1 , X) E(P2 , X)E(P3 , X) ,
(2.5)
AdS
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 [36]) ,
3
∆1 + ∆ 2 + ∆ 3 − d
π h Y C∆i
Γ
.
M3 (P1 , P2 , P3 ) = ig
2
Γ(∆i )
2
(2.6)
i=1
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-tobulk propagator for a vector field can be obtained by applying a differential operator to a
scalar boundary-to-bulk propagator [36]. These operators act as projectors, projecting the
spinning Mellin amplitude MnM1 M2 ···Mn onto a subspace that remains conformally invariant.
Specifically, for a vector field J Mi (Pi ),
b Mi Ai E(Pi , X),
E Mi Ai (Pi , X) = D
(2.7)
b Mi Ai is defined as follows:
where the operator D
b Mi Ai = ∆ i − 1 η Mi Ai + 1 ∂ P Ai .
D
∆i
∆i ∂PiMi i
–5–
(2.8)
b Mi Ai simplifies the index structure of vector
We want to highlight that the operator D
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:
!
n
n
Y
Y
C∆i b Mi A i
Mi Ai
D
.
D
=
(2.9)
Γ(∆i )
i=1
i=1
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.
b Mi Ai can be expressed as ∂A Fδ (Pi ),
An eigenfunction of the differential operator D
i
i
∂Pi
where Fδi (Pi ) denotes any function of Pi with the scaling dimension of δi . That is, Pi ·
∂
∂Pi Fδi (Pi ) = −δi Fδi (Pi ). Then,
b Mi Ai
D
∂
∂PiAi
Fδi (Pi ) =
∆i − 1 − δi ∂
Fδi (Pi ) .
∆i
∂PiMi
Notably, when δi = ∆i − 1, or the scaling dimension of the eigenfunction
(2.10)
∂
A Fδi (Pi )
∂Pi i
is ∆i ,
the eigenvalue is zero.
Q
−γℓm
Let’s see couple of examples. Firstly, by substituting F∆i −1 = f (γij ) ℓ<m Γ(γℓm )Pℓm
for some i (with any function f (γij ) of Mellin variables γij for all j ̸= i) in (2.10), with
P
∆i − 1 = j̸=i γij , we deduce that
0=
Z Y
b Mi Ai
dγℓm D
X
−γik −1
Pk,Ai f (γij )Γ(γik + 1)Pik
=
Z Y
b Mi Ai
dγℓm D
ℓ<m
ℓ<m
(ℓm)̸=(ik)
k̸=i
ℓ<m
−γℓm
Γ(γℓm )Pℓm
Y
X
k̸=i
Pk,Ai f (γij,j̸=k , γik − 1)
Y
−γℓm Γ(γℓm )Pℓm
. (2.11)
ℓ<m
In the final step we have shifted the Mellin variables, γik → γik − 1.4
Q
−γℓm
, for
As another example, by substituting F∆i1 −1 = Pi1 ,Ai2 f (γi1 j ) ℓ<m Γ(γℓm )Pℓm
some i1 and i2 , into (2.10), we deduce that
0=
Z Y
Y
−γℓm
b Mi1 Ai1 ηA A f (γi j )
dγℓm D
Γ(γℓm )Pℓm
1
i1 i2
ℓ<m
ℓ<m
− 2Pi1 ,Ai2
X
k̸=i1
Pk,Ai1 f (γi1 j,j̸=k , γi1 k − 1)
Y
−γℓm Γ(γℓm )Pℓm
. (2.12)
ℓ<m
These identities are crucial for significantly simplifying our target expression for higherpoint functions and uncovering underlying structures.
4
So now
P
j̸=i
γij = ∆i .
–6–
Bulk-to-bulk propagators
A bulk-to-bulk propagator represents the exchange of a primary field with a scaling dimension of ∆, 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,
G∆ (X1 , X2 ) =
Z
i∞
−i∞
dc
2c2
2πi c2 − (∆ − h)2
Z
∂AdS
dQ Eh+c (Q, X1 )Eh−c (Q, X2 ) .
(2.13)
We can deform the integration contour in (2.13) and integrate around the pole, e.g., c =
∆ − h. We subsequently get
Z
G∆ (X1 , X2 ) = (h − ∆)
dQ E∆ (Q, X1 )Ed−∆ (Q, X2 ) .
(2.14)
∂AdS
Similarly, for vector fields, the bulk-to-bulk propagator is [36]
AB
G∆
(X1 , X2 )
Z
i∞
=
−i∞
dc
f∆ (c)
2πi
Z
∂AdS
where
f∆ (c) =
NB
MA
(Q, X2 ) ,
(Q, X1 )ηM N Eh−c
dQ Eh+c
4c2 (h2 − c2 )
(c2 − (∆ − h)2 )2
.
(2.15)
(2.16)
In principle, the existence of second-order poles in f∆ (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 secondorder 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 [36].
For the five-point amplitudes, we have drastically simplified the results from [38] and
expressed them in a more succinct form, as illustrated in (3.25) and (3.35). Subsequently,
–7–
Table 1: The correspondence between gluon flat-space amplitudes and Mellin amplitudes
Description
Minkowski momentum Space
AdS Mellin Space
iki
2Pi
Kinematic variable
Internal propagator
2
Pi
ki ·kj
Three-vertex coupling
g
Four-vertex coupling
g2
1
P
˜ γij
gV3na ,nb ,nc
g 2 V4na ,··· ,nd
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 flatspace 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 V3 and V4 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
π h Qn
Mi Ai should be applied.
i=1 D
2
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:
Z
√
1
SYM = − dd+1 x −g Tr FAB F AB ,
(3.1)
4
a = ∂ Aa − ∂ Aa + gf abc Ab Ac and f abc represent the structure constants of
where FAB
A B
B A
A B
the gauge group. Gluon amplitudes correspond to current correlation functions and have
scaling dimensions ∆i = d − 1.
3.1
Three-point gluon amplitude
The three-point gluon Mellin amplitude shown in Figure 1(a), is [36]5
M1 M2 M3
M3v
π h a1 a2 a3
= ig
f
Γ
2
∆1 + ∆ 2 + ∆ 3 − d + 1
2
where we remind the readers again that
Q3
Mi Ai
i=1 D
=
Y
3
!
Mi Ai
D
IA1 A2 A3 ,
C∆i b Mi Ai
i=1 Γ(∆i ) D
Q3
IA1 A2 A3 = 2ηA1 A2 (P1 − P2 )A3 + cyclic permutations .
5
A detailed calculation can be found in Appendix A
–8–
(3.2)
i=1
and
(3.3)
P2
P1
P3
P2
P3
P2
P3
P1
P4
P1
P4
(a)
(b)
(c)
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.
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,
A3v,A1 A2 A3 = −gf a1 a2 a3 ηA1 A2 (k1 − k2 )A3 + cyclic permutations .
(3.4)
This similarity becomes apparent when the momenta iki,A are mapped to 2Pi,A and the
three-vertex coupling constant g is associated with g V30,0,0 , where
V30,0,0 ≡ Γ(d − 1) .
(3.5)
The three null arguments in (3.5) indicate that the three-vertex is linked to three boundaryto-bulk propagators. This correspondence holds, taking into account the differential operQ
h
ators 3i=1 DMi Ai and a constant factor π2 .
3.2
3.2.1
Four-point gluon amplitudes
Contact diagram
For the four-point contact diagram (Figure 1(b)), the Mellin amplitude is [38]
M1 M2 M3 M4
MContact
!
4
Y
h π
′
′
′
′
= −ig 2
f a1 a4 b f a2 a3 b + f a1 a3 b f a2 a4 b
DMi Ai
2
i=1
!
P4
i=1 ∆i − d
×Γ
ηA1 A2 ηA3 A4 + cyclic perm. of (123) . (3.6)
2
Note that it is also the same as the flat-space counterpart from the Feynman rules in
Yang-Mills theory,
′
′
′
′
AContact = −ig 2 f a1 a4 b f a2 a3 b + f a1 a3 b f a2 a4 b ηA1 A2 ηA3 A4 + cyclic perm. of (123) ,
(3.7)
π h Q4
M
A
i
i
up to 2 i=1 D
, with the identification of the four-vertex coupling constant, i.e.,
0,0,0,0
2
g V4
, where
3d − 4
0,0,0,0
V4
≡Γ
.
(3.8)
2
–9–
3.2.2
Exchange diagram
The s-channel is shown in Figure 1(c). The amplitude can be expressed in terms of the
three-point function by utilizing the factorization in (2.15),
4
DY
J
Mi
E
(Pi )
i=1
Z
Exch
i∞
=
−i∞
dc
f∆ (c)
2πi
Z
M
dQ J M1 (P1 )J M2 (P2 )Jh+c
(Q) ηM N
∂AdS
N
× Jh−c
(Q)J M3 (P3 )J M4 (P4 ) , (3.9)
where the subscripts h ± c of the exchange vector field indicate its scaling dimension. The
integration over Q can be performed by employing Symanzik’s formula [74],




Z
Z Y
n
n
n
n
Y
Y
X
dγij
−γ
dQ
Γ(ℓi ) (−2Pi · Q)−ℓi = π h 
Γ(γij )Pij ij 
δ
γij − ℓi 
2πi
∂AdS
i=1
i<j
i=1
j̸=i
(3.10)
Pn
(note, i=1 ℓi = 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 Pi ’s employing
(2.10) [36, 38]. Therefore, we get
!
2
Ch±c
∆1 + ∆ 2 + h ± c − d + 1
π h a1 a2 b Y Mi Ai
D
Γ
⟨J (P1 )J
= ig f
2
Γ(h ± c)
2
i=1
∆1 + ∆2 − (h ± c) + 1
∆1 − ∆2 + (h ± c) + 1
−∆1 + ∆2 + h ± c − 1 h ± c − 1
×Γ
Γ
Γ
2
2
2
h±c
n
o
∆1 +∆2 −(h±c)+1
∆
−∆
+h±c+1
−∆
+∆
+h±c−1
1
2
1
2
−
M
2
2
2
(−2P1 · Q)−
(−2P2 · Q)−
+ (1 ↔ 2) , (3.11)
× X12
× P12
M1
M2
M
(P2 )Jh±c
(Q)⟩
where
n
o
M
XijM ≡ 2 ηAi Aj PiM − 2δA
P
− (i ↔ j) .
i,A
j
i
(3.12)
Note that in (3.11) we have explicitly performed the action of the differential operator
b M A.
D
h±c
Let’s take a moment to scrutinize the factor h±c−1
h±c in (3.11). One can see that it
possesses simple zeros at c = ±(h + 1). Importantly, with ∆ = d − 1, the simple zeros are
at the same position as the double poles of f∆ (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 [36]
M1 M2 M3 M4
MExch
π h a1 a2 b a3 a4 b
= −g
f
f
2
2
4
Y
!
Mi Ai
D
i=1
n
o
n
o
M Vn,0,0 × Vn,0,0 X
∞
X12
X
34,M
3
3
 , (3.13)
×
d
d
+
n
γ
−
+
n
4n!
Γ
12
2
2
n=0

where (a)n ≡ a(a + 1)(a + 2) · · · (a + n − 1) is the Pochhammer symbol, the contribution of
the boundary points is {XijM } and
d
V3n,0,0 ≡
− n Γ (d − 1)
(3.14)
2
n
– 10 –
stands for the contribution of the three-vertices connecting to one bulk-to-bulk propagator.
Note that with n = 0, V3n,0,0 reduces to the expression defined in (3.5).
The series of simple poles, γ12 = d2 − n, comes from the Gamma function Γ(γ12 ) in
(3.10), where γ12 should be shifted in order to incorporate the power of P12 in (3.11).
Interestingly, for even d, the infinite sum is cut-off at n = d2 . So,
2
2
Γ (d − 1) d2 − 1
Γ (d − 1)
V3n,0,0 × V3n,0,0
+
=
4n!Γ d2 + n γ12 − d2 + n
4Γ d2 γ12 − d2
4Γ d2 + 1 γ12 − d2 + 1
n=0
2
Γ (d − 1) d2 − 1 !
+ ········· +
.
4(γ12 − 1)
∞
X
(3.15)
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
′
AExch = g 2 f a1 a2 b f a3 a4 b
′
i
(ηA1 A2 k1,N − 2ηA1 N k1,A2 − (1 ↔ 2))
(k1 + k2 )2
N
× ηA3 A4 k3N − 2ηA
k
− (3 ↔ 4) , (3.16)
3 3,A4
by replacing the momenta iki → 2Pi , the propagator
i
→
(k1 + k2 )2
4n!Γ
1
d
2
+n
γ12 −
d
2
+n
(3.17)
(with integer n to be summed from 0 to the infinity) and the three-vertex coupling constant
g → g V3n,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 P1 · P3 can be absorbed into the Mellin basis with a shift of Mellin variable
γ13 → γ13 + 1, as in [36]. 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 J M1 (P1 ) · · · J M5 (P5 ) . 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.
– 11 –
P3
P4
P1
P5
P2
Q
Figure 2: Five-point channel consisting of contact and three-point vertex interactions
3.3.1
Channel with a three-vertex and a four-vertex
We initiate our investigation by focusing on a topology, wherein a bulk-to-bulk propagator
establishes a connection between a four-vertex with points P1 , P2 , and P3 , and a threevertex involving points P4 and P5 . This configuration is visually represented in Figure 2.
Using the factorization property given by equation (2.15), the amplitude can be written as:
* 5
Y
+
Z
J Mi (Pi )
i=1
i∞
=
3v4v
−i∞
dc
f∆ (c)
2πi
Z
dQ
∂AdS
* 3
Y
+
M
(Q)
J Mi (Pi )Jh+c
i=1
Contact
N
× ηM N Jh−c
(Q)J M4 (P4 )J M5 (P5 ) . (3.18a)
Let us first look at the expression for contact diagram (3.6). We will identify P4
in (3.6) with Q to match the factorized contact diagram as shown in Figure 2. By acting
b M4 A4 , we substitute P4 with a free index into (3.6). Utilizing
with the differential operator D
Equation (2.10), we can succinctly recasts the four-point contact diagram as follows [38]:
*
4
Y
i=1
+
!
3
Y
′
′
π h a1 a4 b′ a2 a3 b′
C4
DMi Ai
+ f a1 a3 b f a2 a4 b
f
f
2
Γ(∆
4)
i=1
Contact
Z Y
4
′
dγkℓ
∆4 − 1 M4
1 −γ ′
′
×
Γ(γkℓ
)Pkℓ kℓ
ηA3 −
P1,A3 P1M4 + P2M4 + P3M4 + (1 ↔ 2)
2πi
∆4
∆4
k<ℓ
!
P4
i=1 ∆i − d
ηA1 A2 + cyclic perm. of (123) . (3.18b)
×Γ
2
J Mi (Pi )
= −ig 2
P
′
Starting from the constraint imposed by the Mellin variables, denoted as ∆′i ≡ j̸=i γij
M
(note that ∆′i is not necessarily ∆i . For example, the term P1A3 P1 4 has scaling dimension
of −2 for P1 . To compensate that, ∆′1 = ∆1 + 2), we have the opportunity to dispense with
′ to arrive at the following constraint:
γi4
2
3
X
i<j
′
γij
= ∆′1 + ∆′2 + ∆′3 − ∆′4 .
– 12 –
(3.18c)
We now transition to substituting P4 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,
Z i∞
′
′
dc c2 − (h − 1)2
π h a1 bb′ a2 a3 b′
f
f
+ f a1 a3 b f a2 bb f ba4 a5
2
2 2
2
−i∞ 2πi (c − (∆ − h) )
!
5
Y
1
∆4 + ∆5 + h − c − d + 1
∆4 + ∆5 − h + c + 1
×
DMi Ai
Γ
Γ
2Γ(c)Γ(−c)
2
2
i=1
∆4 +∆5 −h+c+1
Γ γ45 −
∆1 + ∆ 2 + ∆ 3 + h + c − d
2
Γ
×
Γ(γ45 )
2
! 3
Z Y
3
3
′
′
′
′
′
′
X ′
h + c − ∆1 − ∆2 − ∆3 Y Γ γij − γij Γ(γij )
dγkℓ
δ
ηA1 A2
×
γkℓ +
2πi
2
Γ(γij )
i<j
k<ℓ
k<ℓ
n
o 2
M
M
M
M
P
+
P
+
P
+
(1
↔
2)
+ cyclic perm. of (123) .
× X45,M × ηA
−
P
1,A
1
2
3
3
3
h+c−1
(3.19)
M1 M2 M3 M4 M5
M3v4v
= g3
Note again that ∆′i depends on the scaling dimension of each term in the last line. In
particular, for a term which has scaling dimension of δi in Pi , ∆′i = ∆i − δi .
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. ∆i = d − 1. The product of the P4,M term
in {X45,M } and the second term in the second curly bracket of (3.19) gives
ηA4 A5 P4,M P1,A3 P1M + P2M + P3M + (1 ↔ 2) → ηA4 A5 P1,A3 (d − 1 − γ45 ) + (1 ↔ 2) ,
(3.20)
−γ14
−γ14 +1
→ γ14 Γ(γ14 )P14
) and
where we have shifted the Mellin variables (e.g. Γ(γ14 )P14
P3
used that i=1 γi4 + γ45 = ∆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 {X45,M }.
The contribution from the multiplication of the P4,A5 term in {X45,M } and the second
term inside the second curly bracket of (3.19) also vanishes due to the antisymmetry under
4 ↔ 5. To see this, we can use (2.12) with
Γ γ45 − ∆4 +∆52−h+c+1
(3.21a)
f (γ45 ) =
,
Γ(γ45 )
and get
f (γ45 )
3
X
i=1
!
Pi,A4
1
P4,A5 = f (γ45 ) ηA4 A5 − f (γ45 − 1)P4,A5 P5,A4 + · · · ,
2
(3.21b)
b M4 A4 . Now it is clear that (3.21b)
where · · · denotes the term vanishing upon the action of D
exhibits symmetry under 4 ↔ 5. Therefore, the antisymmetric property of {X45,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 ∆ = d − 1, the factor
c2 − (h − 1)2
((c2 − (∆ −
h)2 )2
=
– 13 –
c2
1
,
− (h − 1)2
(3.22)
which gives simple poles at c = ±(h − 1). Therefore, with the integration of c around the
pole h − 1, (3.19) can be simplified to:
M1 M2 M3 M4 M5
M3v4v
×
=g
3π
h
2
f
a1 bb′
f
a2 a3 b′
+f
a1 a3 b′
f
a2 bb′
f
5
Y
ba4 a5
!
M i Ai
D
i=1
Γ (γ45 − d + 1)
Γ(γ45 )Γ 1 − d2
Z
Γ (d − 1)
4
!
!
3
3
′
′
′
X
Y
Γ (γkℓ − γkℓ
) Γ(γkℓ
)
dγkℓ
′
δ
γkℓ − d + 1
2πi
Γ(γkℓ )
k<ℓ
k<ℓ
o
n
3d − 4
×Γ
ηA1 A2 X45,A3 + cyclic perm. of (123) .
2
(3.23)
′ , for 1 ≤ k, ℓ ≤ 3, we encounter poles
After integrating (3.23) over the Mellin variables γkℓ
′
at γkℓ = γkℓ + nkℓ , with any non-negative integers nkℓ . However, because of the presence
of a delta function, one of these pole terms does not get integrated out.6 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):
Z
3
′
′
′
Y
Γ (γkℓ − γkℓ
) Γ(γkℓ
)
dγkℓ
2πi
Γ(γkℓ )
!
3
X
δ
k<ℓ
!
′
γkℓ
−d+1
(3.24a)
k<ℓ
′ = γ + n , this term reduces to:
Through the integration around the poles γkℓ
kℓ
kℓ
∞
X
1
i<j γij − d + 1 + m
X
(−1)m
P3
m=0
P3
i<j
nij =m
(γ12 )n12 (γ13 )n13 (γ23 )n23
n12 !n13 !n23 !
(3.24b)
Further simplification leads it to:
∞
X
m
(−1)
γ
−d+1+m
ij
i<j
P
3
i<j
P3
m=0
γij
m
m!
=
∞
X
(−1)m (d − 1 − m)m
P
m!( 3i<j γij − d + 1 + m)
m=0
(3.24c)
P
P
At the pole of (3.24c), 3i<j γij = d − 1 − m, from the constraint j̸=i δij = ∆′i on the
Mellin variables and ∆i = d − 1, we have
γ45 =
∆4 + ∆ 5 + 1 −
P3
i=1
∆i + 2
P3
i<j
γij
2
=
d
−m .
2
(3.24d)
Hence, the first term in the second line of (3.23) becomes
(−1)m d2 − m m
Γ (γ45 − d + 1)
=
.
Γ (γ45 ) Γ 1 − d2
Γ d2 + m
(3.24e)
Finally, the Mellin amplitude becomes remarkably simple:
M1 ···M5
M3v4v
=g
3π
h
2
f
a1 bb′ a2 a3 b′
f
+f
a1 a3 b′ a2 bb′
f
f
ba4 a5
5
Y
!
Mi Ai
D
i=1
n
o
X45,A3 V3m,0,0 × V4m,0,0,0
ηA1 A2 + cyclic perm. of (123) , (3.25)
×
4m!Γ d2 + m (γ45 − d2 + m)
m=0
∞
X
where we have defined
V4m,0,0,0
6
≡ (d − 1 − m)m Γ
Note that the scalar case has a similar construction [37].
– 14 –
3d − 4
2
.
(3.26)
P3
P2
P4
P1
P5
Q
Figure 3: The channel of five-point gluon amplitude for (3.30a)
Note that with m = 0, it reduces to V40,0,0,0 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, d2 }
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,
′
′
′
′
A3v4v = ig 3 f a1 bb f a2 a3 b + f a1 a3 b f a2 bb f ba4 a5 (ηA4 A5 k4,A3 − 2ηA3 A4 k4A5 − (4 ↔ 5))
× η A1 A2
i
+ cyclic perm. of (123) . (3.27)
(k4 + k5 )2
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 iki , it should be replaced by 2Pi . For
the propagator,
i
1
→
,
(3.28)
2
d
(k4 + k5 )
4 γ45 − 2 + m Γ d2 + m m!
with m to be summed over. For the three- and four-vertex coupling constant,
g 7→ g V3m,0,0
and g 2 7→ g 2 V4m,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
In this part of the paper, we focus on the other five-point channel configuration where
there are three-vertices, one links P1 , P2 with a bulk-to-bulk propagator, one links P4 , P5
– 15 –
with another bulk-to-bulk propagator, and the other links P3 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,
* 5
Y
+
Z
J Mi (Pi )
i=1
i∞
=
−i∞
3v3v3v
dc
f∆ (c)
2πi
Z
dQ
∂AdS
×
* 3
Y
+
M
J Mi (Pi )Jh+c
(Q)
i=1
Exch
N
ηM N Jh−c
(Q)J M4 (P4 )J M5 (P5 )
. (3.30a)
The exchange contribution, with ∆i = d − 1, is given by (3.13). Performing the action of
b M4 A4 explicitly, one rewrite the expression as
D
Y
4
J Mi (Pi )
×
4n! Γ
d
2
2
1) ( d2
− n)n
′
+ n γ12
− d2 + n
Γ (d −
n=0
′
Exch
i=1
∞
X
′
= −g 2 f a1 a2 b f a3 a4 b
!
πh
2
3
Y
!
DMi Ai
i=1
Z Y
4
′
dγkℓ
−γ ′
′
Γ(γkℓ
)Pkℓ kℓ
2πi
k<ℓ
2
+
P1M4 + P2M4 + P3M4 HA1 A2 A3 ,
∆4
C4
Γ(∆4 )
∆4 − 1 M4 Ab
η
GA1 A2 A3 Ab
∆4
(3.30b)
where
n
on
o
A ′
GA1 A2 A3 Ab = X12b
Xbb′ ,A3 , Pb ≡ −P1 − P2 − P3 , Pb′ ≡ P1 + P2 ,
(3.30c)
and HA1 A2 A3 is obtained from the product P4Ab GA1 A2 A3 Ab followed by the elimination of
the P4 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,
M1 ···M5
M3v3v3v
3 a1 a2 b′
= −ig f
f
a3 bb′
f
ba4 a5
πh
2
Z
i∞
−i∞
dc c2 − (h − 1)2
2πi (c2 − (∆ − h)2 )2
5
Y
i=1
!
D
1
2Γ(c)Γ(−c)
∆4 +∆5 −h+c+1
Mi Ai
∆4 + ∆ 5 + h − c − d + 1
∆4 + ∆5 − h + c + 1 Γ γ45 −
2
×Γ
Γ
2
2
Γ(γ45 )
!
!
P
Z
3
3
′
′
′
Y
X
h + c − 3i=1 ∆′i
) Γ(γkℓ
)
Γ (γkℓ − γkℓ
dγkℓ
′
δ
γkℓ +
×
2πi
Γ(γkℓ )
2
k<ℓ
k<ℓ
2
∞
n
on
o
X
Γ (d − 1) (− d2 + 1)n
2
M
′
X45
×
GA1 A2 A3 M +
(P1 + P2 + P3 )M HA1 A2 A3 .
d
d
h+c−1
4n!Γ 2 + n γ12 − 2 + n
n=0
(3.31)
Note again that here ∆′i is not necessarily ∆i . From the expression of GA1 A2 A3 M , (3.30c),
P3
P3
P3
P3
′
′
i=1 ∆i =
i=1 ∆i + 2 for the G term, and
i=1 ∆i =
i=1 ∆i + 4 for the H term.
Now we simplify the expression (3.31) further. Taking ∆i = d − 1, the result obtained
M } with the second term inside the second curly
by multiplying ηA4 A5 P4M term from {X45
bracket of (3.31) is (similar to (3.20)), :
2
(3.32)
ηA A (d − 1 − γ45 )HA1 A2 A3 (P1 , P2 , P3 ) ,
h+c−1 4 5
P
where we have used that 3i=1 γi4 + γ45 = ∆4 = d − 1. Since (3.32) is symmetric under
4 ↔ 5, its contribution vanishes due to the antisymmetric property of {X45,M }. The
– 16 –
M }, the second term inside the second curly bracket of (3.31)
product of ηA4 M P4,A5 in {X45
and
Γ γ45 − ∆4 +∆52−h+c+1
(3.33a)
f (γ45 ) =
,
Γ(γ45 )
similar to (3.21b), gives
f (γ45 ) (P1,A4 + P2,A4 + P3,A4 ) P4,A5 HA1 A2 A3 (P1 , P2 , P3 ) =
1
f (γ45 ) ηA4 A5 − f (γ45 − 1)P4,A5 P5,A4 HA1 A2 A3 (P1 , P2 , P3 ) + · · · , (3.33b)
2
b M4 A4 . The expression (3.33b)
where · · · denotes the term vanishing upon the action of D
M },
exhibits symmetry under 4 ↔ 5. As a result, again due to the antisymmetry of {X45
(3.33b) gets canceled out. Consequently, based on the preceding arguments, we can conclude
that the HA1 A2 A3 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.
′ (for 1 ≤ i < j ≤ 3) is at γ +n , with any non-negative integers
Besides, the pole of γij
ij
ij
nij . Integrating over it around the pole, we have
Z Y
3
′
dγkℓ
δ
2πi
k<ℓ
=
∞
X
3
X
!
′
γkℓ
−d
k<ℓ
3
′
′
Y
Γ γij − γij
Γ(γij
)
1
d
′
Γ(γ
γ
−
+n
ij )
12
2
i<j
1
γ
−d+m
ij
i<j
X
(−1)m
P3
m=0
=
∞
X
P3
i<j
nij =m
(γ12 )n12 (γ13 )n13 (γ23 )n23
1
n12 !n13 !n23 !
γ12 + n12 −
m
X
(γ12 )n12 (γ13 + γ23 )m−n12
1
1
(−1)m
n12 !(m − n12 )!
γ12 + n12 −
i<j γij − d + m n12 =0
P3
m=0
d
2
d
2
+n
+n
.
(3.34a)
The above equation has poles at
3
X
i<j
γij = d − m and γ12 =
d
− n − n12 .
2
(3.34b)
Using (3.34b) and shifting n → n − n12 , we get
m
X
(γ12 )n12 (γ13 + γ23 )m−n12
n12 !(m − n12 )!
4n!Γ
n12 =0
2
Γ(d − 1)( d2 − n)n
d
d
2 + n γ12 + n12 − 2 + n
→ V3m,n,0
Γ(d − 1)( d2 − n)n
, (3.34c)
4m!n!Γ d2 + n γ12 − d2 + n
where
min{m,n}
V3m,n,0
( d2 − m + n)m ( d2 − n + n12 )n−n12 (n − n12 + 1)n12
n12 !(m − n12 )!
n12 =0
d
d
=Γ(d − 1)
−n+m
−m+n
.
2
n 2
m
≡Γ(d − 1)m!
X
(3.34d)
Note that V3m,n,0 is explicitly symmetric under m ↔ n. It can also be easily checked that
with n = 0, V3m,0,0 reduces to (3.14). Now, we can rewrite the Mellin amplitude in the
simplified form
– 17 –
M1 ···M5
M3v3v3v
3 a1 a2 b′ a3 bb′ ba4 a5 π
= −ig f
f
5
Y
h
f
!
Mi Ai
D
∞ n
o
X
Xbb′ ,A3 V3m,n,0
2
m,n=0
i=1
n
n
o
o
A ′
Ab
X12b V3n,0,0
X45
V3m,0,0
×
. (3.35)
×
4m!Γ d2 + m γ45 − d2 + m
4n!Γ d2 + n γ12 − d2 + n
For even d, the infinite sums are cut-off at m = d2 and n = d2 , and only finite number of
terms remain.
Just as in the channel (3.25), we can compare (3.35) to its flat-space counterpart,
′
′
A3v3v3v = −g 3 f a1 a2 b f a3 bb f ba4 a5 ηM M ′ qA3 − 2ηM A3 qM ′ − (qM ↔ q ′ M ′ )
i
M
− (4 ↔ 5)
k
× ηA4 A5 k4M − 2ηA
4 4,A5
(k4 + k5 )2
i
′
M′
−
(1
↔
2)
× ηA1 A2 k1M − 2ηA
k
, (3.36)
1,A
2
1
(k1 + k2 )2
with q ≡ −k1 − k2 − k3 and q ′ ≡ k1 + k2 . 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, iki → 2Pi . For the propagator
i
→
(k4 + k5 )2
4m!Γ
d
2
1
+ m γ45 −
d
2
,
+m
d
2
+n
(3.37)
with m to be summed over. And
i
→
(k1 + k2 )2
4n!Γ
1
d
2
+n
γ12 −
,
(3.38)
with n to be summed over. For the three-vertex coupling constant,
g V3n,0,0 ,
g → g V3m,0,0 ,
g V3m,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
4.1
Higher-point gluon amplitudes
Factorization of (n + 1)-point gluon amplitudes
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 P4 and P5 , which are brought
together to a three-vertex, kills the term HA1 A2 A3 . And the remaining term has simple
poles at c = ±(h − 1).
– 18 –
Pn−1
Pn
Pn−2
..
.
P2
h+c
h−c
P1
Pn+1
Q
Figure 4: A (n + 1)-current amplitude for involving a three-vertex
It is important to observe that this simplification does not depend on the details of
the four-point amplitude, i.e. GA1 A2 A3 M and HA1 A2 A3 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.
*n+1
Y
+
J
Mi
(Pi )
Z
i∞
=
−i∞
i=1
dc
f∆ (c)
2πi
Z
dQ
∂AdS
*n−1
Y
+
J
Mi
M
(Pi )Jh+c
(Q)
i=1
N
× ηM N Jh−c
(Q)J Mn (Pn )J Mn+1 (Pn+1 ) . (4.1)
An n-point Mellin amplitude can be written in the following form.
M M1 M2 ···Mn
πh
=
2
n
Y
!
DMi Ai
n
M˜Aa11 aA22···a
···An
i=1
πh
=
2
n−1
Y
!
DMi Ai
i=1
Cn
Γ(∆n )
∆n − 1 Mn An ˜a1 a2 ···an
∂
1
η
MA1 A2 ···An +
H a1 a2 ···an
∆n
∆n ∂PnMn A1 A2 ···An−1
, (4.2)
An ˜a1 a2 ···an
n
where HAa11Aa22···a
···An−1 ≡ Pn MA1 A2 ···An .
As in the examples (2.11), (2.12) and their generalizations with more free indices, we
can always use (2.10) to replace PnA by the other PiA ’s. In this way, we can eliminate the
dependences of M˜A1 A2 ···An and HA1 A2 ···An−1 on Pn . Then, the second term in the last line
of (4.2) can be further calculated as follows.
∂
HA1 A2 ···An−1 (P1 , P2 , · · · , Pn−1 )
Mn
∂Pn
2
n−1
X
i=1
n
Y
′
−γkℓ
′
Γ(γkℓ
)Pkℓ
=
k<ℓ
′
′
PiMn HA1 A2 ···An−1 (γin
→ γin
− 1)
– 19 –
n
Y
k<ℓ
′
−γkℓ
′
Γ(γkℓ
)Pkℓ
, (4.3)
′ denote the Mellin variables for the lower point Mellin amplitude M . Plugging
where γij
n
(3.11), (4.2) and (4.3) in (4.1), we have
πh
2
Z
i∞
dc c2 − (h − 1)2
2πi (c2 − (∆ − h)2 )2
n+1
Y
!
1
2Γ(c)Γ(−c)
i=1
∆n +∆n+1 −h+c+1
Γ γ
−
n(n+1)
2
∆n + ∆n+1 + h − c − d + 1
∆n + ∆n+1 − h + c + 1
×Γ
Γ
2
2
Γ(γn(n+1) )
!
P
Z n−1
n−1
′
′
′
′
Y dγij
X ′
Γ γij − γij
Γ(γij
)
h + c − n−1
i=1 ∆i
×
δ
γkℓ +
2πi
Γ(γij )
2
i<j
k<ℓ
(
)
n−1
o
n
X
2
a1 a2 ···an−1 b
a1 a2 ···an−1 b
M
′
′
˜
× Xn(n+1)
MA1 A2 ···An−1 M (γij ) +
Pi,M HA1 A2 ···An−1 (γij ) . (4.4)
h + c − 1 i=1
M M1 ···Mn = igf an an+1 b
−i∞
DMi Ai
Again ∆′i ≡ ∆i − δi where δi denotes the scaling dimension of Pi 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
P
∆′i .
dimension n−1
i
Now we take ∆i = d−1 and show that the contribution of HA1 A2 ···An−1 vanishes. First,
ηAn An+1 PnM
n−1
X
i=1
1
Pi,M HA1 A2 ···An−1 → − ηAn An+1 (d − 1 − γn(n+1) )HA1 A2 ···An−1 , (4.5)
2
where we have shifted the Mellin variables γin → γin + 1 (no prime) for each term in the
P
sum and used that n−1
i=1 γin + γn(n+1) = ∆n = d − 1. Since (4.5) is symmetric under the
interchange of labels n and n + 1 (i.e., n ↔ n + 1), the contribution of it vanishes due to
M
the antisymmetry of {Xn(n+1
}. Second, using (2.12) with
f (γn(n+1) ) =
Γ γn(n+1) −
∆n +∆n+1 −h+c+1
2
Γ(γn(n+1) )
,
(4.6a)
we can get
f (γn(n+1) )
n−1
X
Pi,An Pn,An+1 HA1 A2 ···An−1
i=1
=
1
f (γn(n+1) )ηAn An+1 − f (γn(n+1) − 1)Pn,An+1 Pn+1,An HA1 A2 ···An−1 + · · · , (4.6b)
2
b Mn An . The expression (4.6b)
where · · · denotes the term vanishing upon the action of D
again exhibits symmetry under n ↔ n + 1. As a result, its contribution gets completely
M
canceled out from the antisymmetry of {Xn(n+1
}. Combining these, we can conclude that
the H term in (4.4) vanishes.
Finally, the integration around the simple pole c = ∆−h (with ∆ = d−1) and the poles
′
γij = γij + nij with any non-negative integers nij gives the important recursion formula
from an n-point amplitude to an (n + 1)-point amplitude.
– 20 –
P6
P5
Q
P1
P4
P2
P3
Figure 5: The “snowflake” channel of six point gluon amplitude for (4.8)
M M2 ···Mn+1
1
Mn+1
×
=
2
X
Pn−1
i<j
n+1
Y
πh
!
DMi Ai
igf an an+1 b
∞
X
m=0
i=1
n−1
Y
4Γ
d
2
n
o
M
Xn(n+1)
V3m,0,0
+ m (γn(n+1) − d2 + m)
(γij )nij ˜a1 a2 ···an−1 b
Mn,A1 A2 ···An−1 M (P1 , P2 , · · · , Pn−1 )
nij !
nij =m i<j
′ →γ +n
γij
ij
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
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,
M1 M2 ···M6
MSnowflake
πh
=
2
×
6
Y
D
∞
n
oX
M
igf a5 a6 b X56
m=0
i=1
X
P4
!
Mi Ai
4
Y
(γij )nij
i<j
i<j nij =m
nij !
4Γ
d
2
V3m,0,0
+ m (γ56 −
a1 a2 a3 a4 b
M˜3v3v3v,A
(P1 , P2 , P3 , P4 )
1 A2 ···A4 M
d
2
+ m)
′ →γ +n
γij
ij
ij
. (4.8)
In (4.8) M˜3v3v3v can achieved by eliminating the P3 dependence in the Mellin amplitude
(3.35), interchanging the labels 1 ↔ 2 and 3 ↔ 5, and replacing (A5 , a5 ) by (M, b). We
first eliminate P3 in the Mellin amplitude. One appearance of P3 is either in the inner
products Pi · P3 − (i ↔ j) (where (i, j) = (1, 2) or (4, 5)), or in the terms with free indices,
Pi,Aj P3,Ai − (i ↔ j). For the former, by shifting the Mellin variable γi(j)3 → γi(j)3 + 1 we
– 21 –
have
X
1
1 X
Pi · P3 − (i ↔ j) → − γi3 − (i ↔ j) =
γik − (i ↔ j) → −
Pi · Pk − (i ↔ j) ,
2
2
k̸=i,3
k̸=i
(4.9)
where in the second step we have used ∆i = ∆j = d − 1 and in the last step we have shifted
γi(j)k → γi(j)k − 1. For the latter, we can use (2.12) to replace P3,Ai(j) . So, it becomes
X
Pi,Aj P3,Ai − (i ↔ j) → −Pi,Aj
Pk,Ai − (i ↔ j) .
(4.10)
k̸=3
Combining (4.9) and (4.10), the net result is that we can replace P3 in (3.35) by
P
− k̸=3 Pk . After getting rid of the P3 dependence, and interchanging the labels 1 ↔ 2
and 3 ↔ 5, we can read off that
′′ ′
′′
′
a1 a2 ···a5
= −ig 3 f a1 a2 b f a5 b b f a3 a4 b
M˜3v3v3v,A
1 A2 ···A5
∞ n
o
X
′
Xb′′ b′ ,A5 V3m ,n,0
m′ ,n=0
n
n
o
o
′
A ′
A ′′
X12b V3n,0,0
X34b V3m ,0,0
′
×
′
, (4.11)
×
4m′ !Γ d2 + m′ γ34
4n!Γ d2 + n γ12
− d2 + m′
− d2 + n
with Pb′′ ≡ P3 + P4 , Pb′ ≡ P1 + P2 . Plugging it in (4.8), we see that the poles are at
γ56 =
d
− m,
2
γ12 =
d
− n − n12 ,
2
γ34 =
d
− m′ − n34 .
2
(4.12a)
And from the delta function restriction,
P4
∆i − ∆5 − ∆6 + 2
γ13 + γ14 + γ23 + γ24 = i=1
+ γ56 − γ12 − γ34
2
d
= − m + n + m′ + n12 + n34 .
2
(4.12b)
The sum over nij in (4.8) at the poles yields
X
P4
i<j
4
Y
(γij )nij
nij =m i<j
nij !
=
m
X
m−n
X12
n12 =0 n34 =0
( d2 − n − n12 )n12 ( d2 − m′ − n34 )n34
n12 !
n34 !
(4.12c)
( d − m + n + m′ + n12 + n34 )m−n12 −n34
.
× 2
(m − n12 − n34 )!
Shifts of m′ → m′ − n34 and n → n − n12 lead to
m
X
m−n
X12
n12 =0 n34 =0
×
( d2 − n − n12 )n12 ( d2 − m′ − n34 )n34 ( d2 − m + n + m′ + n12 + n34 )m−n12 −n34
n12 !
n34 !
(m − n12 − n34 )!
Γ(d − 1)( d2 − m′ )m′
4m′ !Γ d2 + m′ γ34 + n34 −
→
′
d
2
+ m′
V3m ,n,0
4n!Γ
d
2
Γ(d − 1)( d2 − n)n
+ n γ12 + n12 −
d
2
+n
Γ(d − 1)( d2 − m′ )m′
Γ(d − 1)( d2 − n)n
1
m′ ,n,m
, (4.12d)
V
3
m! 4m′ !Γ d2 + m′ γ34 − d2 + m′
4n!Γ d2 + n γ12 − d2 + n
– 22 –
where we have defined
m′ ,n,m
V3
min{m,n} min{m−n12 ,m′ }
m!
n12 !n34 !
d
2
− m + n + m′
m−n12 −n34
(m′ − n34 + 1)n34
(m − n12 − n34 )!
n34 =0
n12 =0
′
d
d
′
×
+ m − n34
(n − n12 + 1)n12
+ n − n12
V3m −n34 ,n−n12 ,0 . (4.12e)
2
2
n34
n12
≡
X
X
Presumably this definition is symmetric among m′ , 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 V3m ,n,0 , consistent with the l.h.s. Besides, from
′
′
′
the definition (4.12e), we find V3m ,0,m = V30,m ,m , and it is equal to V3m,m ,0 from the
definition in (3.34d).
′
With the aid of the newly defined V3m ,n,m , we can rewrite the Mellin amplitude (4.8)
as
M1 M2 ···M6
MSnowflake
b′
b′′
= g 4 f a1 a2 f a3 a4 f a5 a6 b f
∞
X
m′ ,n,m=0
bb′′ b′
πh
2
6
Y
!
DMi Ai
i=1
o
n
M Vm,0,0
X56
3
d
4m!Γ 2 + m γ56 − d2 + m
n
o
A ′
X12b V3n,0,0
×
. (4.13)
+ m′
4n!Γ d2 + n γ12 − d2 + n
n
o
′
Xb′′ b′ ,M V3m ,n,m ×
n
o
′
A ′′
X34b V3m ,0,0
×
4m′ !Γ d2 + m′ γ34 − d2
Compare it with the flat-space analog
′
′′
′′ ′
M1 M2 ···M6
ASnowflake
= g 4 f a1 a2 b f a3 a4 b f a5 a6 b f bb
× ηA5 A6 k5M
′′
′′
′′
′′
(ηM ′′ M ′ qM
− 2ηM ′′ M qM
↔ q ′ M ′ ))
′ − (q M
i
M
M ′′
M ′′
− 2ηA
k
η
−
(5
↔
6)
k
−
2η
k
−
(3
↔
4)
A3 A4 3
A3 3,A4
5 5,A6
(k5 + k6 )2
i
i
M′
M′
η
k
−
2η
k
−
(1
↔
2)
, (4.14)
×
A
A
1,A
1
A
1
2
2
1
2
(k3 + k4 )
(k1 + k2 )2
b
where q ′′ ≡ k3 +k4 and q ′ ≡ k1 +k2 , we have the dictionary that iki → 2Pi for the momenta,
i
→
(ki + kj )2
4m!Γ
d
2
1
+ m γij −
d
2
+m
,
(4.15)
′
with m to be summed over for the propagators, and g → g V3m ,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. Using (3.11), (3.25) and (4.7), we have
– 23 –
P6
P5
Q
P1
P4
P3
P2
Figure 6: The channel of six point gluon amplitude for (4.16)
M1 M2 ···M6
M3v3v4v
= ig 4
P4
i<j
×
f a1 b
nij =m i<j
′′ ′
b
∞
X
!
DMi Ai
m=0
i=1
4
Y
X
×
6
Y
πh
2
(γij )nij
nij !
′
f a2 bb + f a2 b
′′ ′
b
4Γ
d
2
∞
X
n=0
d
2
4n!Γ
f a1 bb
n
′
V3m,0,0
+ m (γ56 −
d
2
+ m)
V3n,0,0 × V4n,0,0,0
+ n (γ34 + n34 −
d
2
+ n)
n
o
′′
M
f a5 a6 b X56
´f a3 a4 b
o
X34,M ηA1 A2 + cyclic perm. of (A1 a1 , A2 a2 , M b) .
(4.16)
The poles of Mellin variables are at
d
d
(4.17a)
− m, γ34 = − n − n34 .
2
2
Then, from the equation which results from the restrictions on the Mellin variables, i.e.,
γ56 =
∆5 + ∆6 + 1 − 2γ56 =
4
X
i=1
∆i + 1 − 2
4
X
γij ,
(4.17b)
i<j
we have
X
4
Y
(γij )nij
nij !
P4
i<j
i<j nij =m
=
m
X
( d2 − n − n34 )n34 (d − 1 − m + n + n34 )m−n34
.
n34 !(m − n34 )!
(4.17c)
n34 =0
A shift of n → n − n34 amounts to
m
X
( d2 − n − n34 )n34 (d − 1 − m + n + n34 )m−n34
n34 !(m − n34 )!
n!Γ
n34 =0
d
Vm,n,0,0
2 −n n
→ 43d−4 ,
d
Γ 2 m!n!Γ 2 + n (γ34 − d2 + n)
d
2
d
2
− n n (d − 1 − n)n
+ n (γ34 + n34 − d2 + n)
(4.17d)
where
3d − 4
2
min{m,n}
(d − 1 − m + n)m−n34
n34 !(m − n34 )!
n34 =0
d
× (n − n34 + 1)n34
+ n − n34
(d − 1 − n + n34 )n−n34 .
2
n34
V4m,n,0,0 ≡Γ
m!
X
– 24 –
(4.17e)
P5
P4
P3
P6
P7
P2
Q
P1
Figure 7: The "scarecrow" channel of seven-point gluon amplitude for (4.20)
One can check that from this definition V4m,0,0,0 = V40,m,0,0 , which is also identical to the
V4m,0,0,0 given in (3.26). Now, we can rewrite (4.16) as
M1 M2 ···M6
M3v3v4v
= ig 4
πh
2
6
Y
!
DMi Ai
i=1
o
n
M Vm,0,0
X56
3
V4m,n,0,0 ×
d
4m!Γ 2 + m (γ56 − d2 + m)
m,n=0
∞
X
n
o
M ′′ Vn,0,0
X34
3
a5 a6 b a3 a4 b′′
a1 b′′ b′ a2 bb′
a2 b′′ b′ a1 bb′
×
f
f
f
f
+
f
f
4n!Γ d2 + n (γ34 − d2 + n)
× ηA1 A2 ηM M ′′ + cyclic perm. of (A1 a1 , A2 a2 , M b)) . (4.18)
Like the previous cases, the Mellin amplitude (4.18) can be obtained from the flat-space
Feynman rules by replacing iki → 2Pi for the momenta,
i
→
(ki + kj )2
4m!Γ
d
2
1
+ m γij −
d
2
+m
,
(4.19)
with m to be summed over for the propagators, g → g V3m,0,0 for the three-vertex coupling
constant and g 2 → g 2 V4m,n,0,0 for the four-vertex coupling constant.
4.4
Seven-point amplitude: scarecrow channel
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
M1 M2 ···M7
MScarecrow
= −g
5π
h
2
X
×
P5
i<j
nij =m
7
Y
!
Mi Ai
D
i=1
5
Y
(γij )nij
nij !
i<j
∞
X
m=0
Γ (d − 1) d2 − m m
4Γ d2 + m (γ67 − d2 + m)
∞
X
m′ ,n=0
V4m
′
,n,0,0
Γ (d − 1) d2 − m′ m′
4m′ !Γ d2 + m′ (γ45 + n45 − d2 + m′ )
n
o
n
o
Γ (d − 1) d2 − n n
′′
M
N
f ba6 a7 X67
f a4 a5 c X45
f a2 a3 b
d
d
4n!Γ 2 + n (γ23 + n23 − 2 + n)
′′ ′
n
o
′
′′ ′
′
× f bb b f a1 cb + f a1 b b f bcb
X23,N ηM A1 + cyclic perm. of (M b, A1 a1 , N c) .
×
– 25 –
(4.20)
At the pole,
d
d
d
− m, γ23 = − n − n23 , γ45 = − m′ − n45 ,
2
2
2
P5
P5
with ∆6 + ∆7 + 1 − 2γ67 = i=1 ∆i + 2 − 2 i<j γij , we have
γ67 =
X
P5
i<j
=
m
X
(4.21a)
5
Y
(γij )nij
nij =m i<j
m−n
X23
n23 =0 n45 =0
nij !
( d2 − n − n23 )n23 ( d2 − m′ − n45 )n45 (d − 1 − m + n + m′ + n23 + n45 )m−n23 −n45
.
n23 !n45 !(m − n23 − n45 )!
(4.21b)
Shift n → n − n23 and
m
X
m′
→
m′
− n45
m−n
X23
( d2 − n − n23 )n23 ( d2 − m′ − n45 )n45 (d − 1 − m + n + m′ + n23 + n45 )m−n23 −n45
n23 !n45 !(m − n23 − n45 )!
n23 =0 n45 =0
d
d
′
m′ ,n,0,0
2 − m m′
2 −n n
×
V4
m′ !Γ d2 + m′ (γ45 + n45 − d2 + m′ )
n!Γ d2 + n (γ23 + n23 − d2 + n)
d
d
′
m′ ,n,m,0 1
2 − m m′
2 −n n
, (4.21c)
→ V4
m! m′ !Γ d2 + m′ (γ45 − d2 + m′ ) n!Γ d2 + n (γ23 − d2 + n)
where
min{m,n} min{m−n23 ,m′ }
(d − 1 − m + n + m′ )m−n23 −n45 ′
(m − n45 + 1)n45
n23 !n45 !(m − n23 − n45 )!
n45 =0
n23 =0
′
d
d
+ m′ − n45
+ n − n23
×
V4m −n45 ,n−n23 ,0,0 × (n − n23 + 1)n23
. (4.21d)
2
2
n45
n23
′
V4m ,n,m,0
≡ m!
X
X
One can check that from this definition V4m,n,0,0 = V40,m,n,0 = V4m,0,n,0 and reduces to
V4m,n,0,0 in (4.17e). Furthermore, with (4.21d), we can rewrite the Mellin amplitude (4.20)
as
n
o
M Vm,0,0
X
67
3
′
M1 M2 ···M7
MScarecrow
= −g 5
DMi Ai
V4m ,n,m,0
d
2
4m!Γ 2 + m (γ67 − d2 + m)
i=1
m,n,m′ =0
n
o
n
o
′
N Vn,0,0
M ′ Vm ,0,0
X45
X
23
3
3
′′
×
×
f a6 a7 b f a4 a5 c f a2 a3 b
d
d
d
d
′
′
′
4m !Γ + m (γ45 − 2 + m ) 4n!Γ 2 + n (γ23 − 2 + n)
′′ ′ 2 ′
′′ ′
′
× f bb b f a1 cb + f a1 b b f bcb ηM A1 ηM ′ N + cyclic perm. of (M b, A1 a1 , M ′ c) , (4.22)
πh
7
Y
!
∞
X
which can be mapped from the flat-space counterpart by replacing iki → 2Pi for the momenta,
i
1
,
→
(4.23)
2
d
(ki + kj )
4m!Γ 2 + m γij − d2 + m
with m to be summed over for the propagators, g → g V3m,0,0 for the three-vertex coupling
′
constant and g 2 → g 2 V4m ,n,m,0 for the four-vertex coupling constant.
– 26 –
P3
P4
P2
P5
P6
P1
Q
P7
P8
Figure 8: The “drone” channel of eight-point gluon amplitude for (4.24)
4.5
Eight-point amplitude: drone channel
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
M1 M2 ···M8
MDrone
∞
X
= −ig
6π
8
Y
h
2
!
Mi Ai
D
i=1
∞
X
m=0
Γ (d − 1) d2 − m m
4m!Γ d2 + m (γ78 − d2 + m)
X
P6
i<j
nij =m
6
Y
(γij )nij
nij !
i<j
Γ (d − 1)
Γ (d − 1) d2 − m′ m′
4n′ !Γ d2 + n′ (γ56 + n56 − d2 + n′ ) 4m′ !Γ d2 + m′ (γ34 + n34 − d2 + m′ )
n′ ,n,m′ =0
o
o
n
o
n
n
Γ (d − 1) d2 − n n
′
N
M′
M
f a3 a4 c X34
f a5 a6 c X56
×
f a7 a8 b X78
d
d
4n!Γ
+ n (γ12 + n12 − 2 + n)
n
o
′ ′′2 ′
′ ′
′′ ′
c b b bcb′
X12,N ηM M ′ + cyclic perm. of (M ′ c′ , M b, N c) . (4.24)
+ f bb b f c cb
f
× f
d
2
′
′
V4m ,n,n ,0
− n′ n′
At the pole,
γ78 =
d
− m,
2
γ12 =
d
− n − n12 ,
2
γ34 =
d
− m′ − n34 ,
2
γ56 =
d
− n′′ − n56 , (4.25a)
2
with
∆7 + ∆8 + 1 − 2γ78 =
6
X
i=1
∆i + 3 − 2
6
X
γij ,
(4.25b)
i<j
we have
X
P6
i<j
=
m
X
6
Y
(γij )nij
nij =m i<j
m−n
12 −n34
X12 m−nX
n12 =0 n34 =0
×
nij !
n56 =0
( d2 − n − n12 )n12 ( d2 − m′ − n34 )n34 ( d2 − n′ − n56 )n56
n12 !n34 !n56 !
(d − 1 − m + n + m′ + n′ + n12 + n34 + n56 )m−n12 −n34 −n56
.
(m − n12 − n34 − n56 )!
– 27 –
(4.25c)
Shift n → n − n12 , m′ → m′ − n34 and n′ → n′ − n56
m
X
m−n12 m−n12 −n34
X
n12 =0 n34 =0
X
n56 =0
( d2 − n − n12 )n12 ( d2 − m′ − n34 )n34 ( d2 − n′ − n56 )n56
n12 !n34 !n56 !
d
− n′ n′
(d − 1 − m + n + m′ + n′ + n12 + n34 + n56 )m−n12 −n34 −n56 m′ ,n,n′ ,0
2
×
V4
(m − n12 − n34 − n56 )!
n′ !Γ d2 + n′ (γ56 + n56 − d2 + n′ )
!
d
d
− m′ m′
−n n
2
2
× ′
m !Γ d2 + m′ (γ34 + n34 − d2 + m′ ) n!Γ d2 + n (γ12 + n12 − d2 + n)
!
d
d
d
− n′ n′
− m′ m′
−n n
m′ ,n,n′ ,m
2
2
2
,
→ V4
n′ !Γ d2 + n′ (γ56 − d2 + n′ ) m′ !Γ d2 + m′ (γ34 − d2 + m′ ) n!Γ d2 + n (γ12 − d2 + n)
(4.25d)
where
′
′
V4m ,n,n ,m
min{m,n} min{m−n34 ,m′ } min{m−n12 −n34 ,n′ }
(d − 1 − m + n + m′ + n′ )m−n12 −n34 −n56
n12 !n34 !n56 !(m − n12 − n34 − n56 )!
n12 =0
n34 =0
n56 =0
′
′
d
× V4m −n34 ,n−n12 ,n −n56 ,0 (n′ − n56 + 1)n56
+ n′ − n56
2
n56
d
d
× (m′ − n34 + 1)n34
+ m′ − n34
(n − n12 + 1)n12
+ n − n12
. (4.25e)
2
2
n34
n12
≡ m!
X
X
X
′
′
′
′
One can check that from this definition V4m ,n,m,0 = V4m ,n,0,m = V4m ,0,n,m = V40,m ,n,m and
′
′
′
reduces to V4m ,n,m,0 in (4.21d). With such a definition of V4m ,n,n ,m , we can rewrite the
Mellin amplitude (4.24) as
M1 M2 ···M8
MDrone
= −ig 6
πh
8
Y
n
o
M Vm,0,0
X
78
3
′
′
V4m ,n,n ,m
d
4m!Γ 2 + m (γ78 − d2 + m)
n′ ,n,m′ ,m=0
n
o
′
N Vm ,0,0
X34
3
×
+ n′ ) 4m′ !Γ d2 + m′ (γ34 − d2 + m′ )
∞
X
!
DMi Ai
2
i=1
n
o ′
M ′ Vn ,0,0
X56
3
×
d
4n′ !Γ 2 + n′ (γ56 − d2
n
o
M ′′ Vn,0,0
X12
3
′
′′
×
f a7 a8 b f a5 a6 c f a3 a4 c f a1 a2 b
d
d
4n!Γ 2 + n (γ12 − 2 + n)
′ ′′ ′
c b b bcb′
bb′′ b′ c′ cb′
′ ′
′
′′
× f
f
+f
f
ηM M ηN M + cyclic perm. of (M c , M b, N c) , (4.26)
which is related to the flat-space counterpart by the following replacements up to an overall
π h Q8
Mi Ai . For the momenta, ik → 2P . For propagators,
i
i
i=1 D
2
i
→
(ki + kj )2
4m!Γ
d
2
1
+ m γij −
d
2
+m
,
(4.27)
with m to be summed over. For the three-vertex coupling constant, g → g V3m,0,0 , and for
′
′
the four-vertex coupling constant g 2 → g 2 V4m ,n,n ,m .
– 28 –
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 ∂xA to the
Fourier basis eiki x introduces a factor, ik A . In contrast, within AdS space, the operation of
Mi Ai
∂X A on the boundary-to-bulk propagator, represented as E∆
(Pi , X), produces a factor
i
of 2PiA . This analogy provides a rationale for the presence of ik and 2Pi on respective sides
of the established dictionary.
Indeed, the map 2Pi ↔ iki 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.
P
P
For the internal propagator in the flat-space amplitude, i.e. i/( i ki )2 = i/2 i<j ki ·kj ,
P
we have 1/ ˜
γij on the Mellin side, where we have defined
i<j
X
˜
i<j
γij ≡ 4m! Γ


X
X
(d − 1) 1
d
+m 
−
γij +
(d − 1 − δi ) + m ,
2
2
2
i<j
(4.28)
i
where again δi denotes the scaling dimension of Pi in the Mellin amplitude. The map
between the Mandelstam invariants, ki · kj and γij resembles the relationship in the scalar
scenario, as referenced in [36].
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 ∞. Then, for a three-vertex connecting the propagators
associated with integers m1 , m2 and m3 , the three-vertex coupling is
g ↔ g V3m1 ,m2 ,m3 .
(4.29)
This may contain boundary-to-bulk propagators with mi = 0. Similarly, for a four-vertex
connecting propagators associated with integers m1 , m2 , m3 and m4 , the coupling is supposed to be
(4.30)
g 2 ↔ g 2 V4m1 ,m2 ,m3 ,m4 .
It is also noteworthy that in scalar cases a high energy limit γij → ∞ takes Mellin
amplitudes to flat-space amplitudes up to a transform [37]. Specifically, as reviewed in
Appendix C.1, in the flat-space limit an n-point scalar Mellin amplitude becomes
Z ∞
n
1 Pn
π h Y C∆i
Mn (∆i , γij ) ≈
dββ 2 ( i=1 ∆i −d)−1 e−β An (pi · pj = 2βγij ) .
2
Γ(∆i ) 0
i=1
– 29 –
(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
!Z
Pn
n
∞
π h Y Mi Ai
i=1 ∆i −d −1 −β
M1 ···Mn
2
Mn
≈
e
D
dββ
2
0
i=1
(4.32)
p
i
1
P
× An,A1 A2 ···An
, iki → 2 βPi .
P 2 →
4β i<j γij
ki
i
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.7
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 [10, 76–78]. 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 [14, 79–93]. 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 [12, 94, 95]). Previous investigations into
loop calculations in flat space have revealed crucial links between trees and loops, bridging
7
For a bispinor formalism of AdS correlators in embedding space, see [75]
– 30 –
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.
A
Three-point scalar and gluon amplitudes: Schwinger trick
In this appendix, we review the calculation of the three-point AdS amplitudes [36]. For the
scalar case, as discussed above, the correlation function can be calculated by integrating
the bulk-to-boundary propagators, i.e.
⟨O1 (P1 )O2 (P2 )O3 (P3 )⟩ = ig
Z
AdS
dX E(P1 , X) E(P2 , X)E(P3 , X) .
(A.1)
To perform the integration, it is convenient to express the bulk-to-boundary propagator
with Schwinger parameter,
C∆i
E(Pi , X) =
Γ(∆i )
Z
+∞
0
dti ∆i 2ti Pi ·X
t e
.
ti i
(A.2)
Then, the integration becomes
⟨O1 (P1 )O2 (P2 )O3 (P3 )⟩ = ig
Z +∞
Z
3
Y
dti ∆i
C∆i
dXe2T ·X ,
t
Γ(∆i ) 0
ti
AdS
(A.3)
i=1
P
where T ≡ 3i=1 ti Pi . Since ti are positive and Pi are null vectors, T must be time-like. In
the rest frame where T = (T 0 , T µ ) = (|T |, 0), parametrize the AdSd+1 space by
X=
1 + x20 + x2 1 − x20 − x2 µ
,
,x
2x0
2x0
,
(A.4)
which satisfies the equation (2.1) for R = 1. Then,
Z
dXe
AdS
2T ·X
Z +∞
2
1+x2
0 +x |T |
dx0
d −
x0
=
d
xe
xd+1
0
0
0
Z +∞
dx0 −x0 + Tx 2
0 .
=π h
e
xh+1
0
0
Z
+∞
– 31 –
(A.5)
So,
Z +∞
3
Y
C∆i
Γ(∆i ) 0
i=1
Z +∞
3
Y
C∆i
h
=igπ
Γ(∆i ) 0
⟨O1 (P1 )O2 (P2 )O3 (P3 )⟩ =igπ h
i=1
=igπ h Γ
P3
i=1 ∆i
2
−d
dti ∆i
t
ti
Z
+∞
0
dti ∆i T 2
t e
ti
!
dx0 −x0 + Tx 2
0
e
xh+1
0
Z
+∞
0
3
Y
C∆i
Γ(∆i )
dx0
P3
∆i
h− i=1
+1
2
e−x0
x0
Z
i=1
+∞
0
dti ∆i −t1 t2 P12 −t1 t3 P13 −t2 t3 P23
t e
,
ti i
(A.6)
where Pij ≡ −2Pi · Pj .
For the integration over Schwinger parameters ti , we change the variables mij = ti tj .
Then,
! 3
P3
3 Z +∞
Y C∆ Y
dmij γij −mij Pij
πh
i
i=1 ∆i − d
m e
⟨O1 (P1 )O2 (P2 )O3 (P3 )⟩ =ig Γ
2
2
Γ(∆i )
mij ij
0
i=1
i<j
! 3
P3
3
h
Y C∆ Y
π
−γ
i
i=1 ∆i − d
=ig Γ
Γ(γij )Pij ij ,
2
2
Γ(∆i )
i=1
i<j
(A.7)
which gives the three-point scalar Mellin amplitude (2.6).
Similarly, for the three-point gluon amplitude, we have
⟨J1 (P1 )J2 (P2 )J3 (P3 )⟩
Z
a1 a2 a3
= −igf
dX ηA1 A2
AdS
= −igf
=
3 Z
Y
+∞
Z
dti ∆1 ∆2 +1 ∆3
t t
t
dXe2T ·X
D
2ηA1 A2 P2,A3
ti
AdS
0
i=1
i=1
i
− (1 ↔ 2) + cyclic permutations
!
P3
3
3
Y
Y
−γ
i=1 ∆i − d + 1
igf a1 a2 a3
DMi Ai IA1 A2 A3
Γ(γij )Pij ij , (A.8)
2
a1 a2 a3
πh
Γ
2
3
Y
∂
M2 A2
E
(P2 , X) E M2 A2 (P2 , X)E M3 A3 (P3 , X)
∂XA3
− (1 ↔ 2) + cyclic permutations
Mi Ai
h
i=1
i<j
from which the Mellin amplitude, (3.2), can be read off.
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
Z
Z
n
n Z +∞
Y
Y
dti ℓi
−ℓi
dQ
Γ(ℓi ) (−2Pi · Q)
=
t
dQe2T ·Q ,
(B.1)
t
i
∂AdS
0
∂AdS
i=1
i=1
– 32 –
P
where T ≡ ni=1 ti Pi . Since ti are positive and Pi are null vectors, T must be time-like. In
the rest frame where T = (T 0 , T µ ) = (|T |, 0), parametrize the boundary of AdSd+1 by
2
x + 1 x2 − 1 µ
(B.2)
Q=
.
,
,x
2
2
Then,
Z
dQe2T ·Q =
∂AdS
where |T | =
q P
− ni,j=1 ti tj Pi · Pj .
π h −|T |
e
,
|T |h
(B.3)
Change variables ti → ti |T |. Then,
Pn
Z +∞
Z
n Z +∞
n
Y
det(∂ti tj |T |) |T | i=1 ℓi Y dti ℓi −|T |2
dti ℓi
2T ·Q
h
t
dQe
=π
t e
.
ti
|T |n
ti
|T |d
0
∂AdS
0
i=1
(B.4)
i=1
The Jacobian is
Pn
P
2 nk=1 tk Pk · Pi
det(∂ti tj |T |) = det δij |T | − tj
= |T |n
2|T |
k,i=1 tk ti Pk
|T |2
1−
where in the second step we have used the formula det(δij + Ai Bj ) = 1 +
Pn
i=1 ℓi = d, (B.4) becomes
2π
h
Z
n
+∞ Y
0
i=1
dti ℓi −|T |2
t e
= 2π h
ti
Z
0
n
+∞ Y
i=1
· Pi
!
= 2|T |n ,
(B.5)
i Ai Bi . For
P
dti ℓi − Pni<j ti tj Pij
t e
.
ti
(B.6)
Recall that in (A.7) we have derived that
2π
h
3 Z
Y
i=1
+∞
0
3
Y
dti ∆i −t1 t2 P12 −t1 t3 P13 −t2 t3 P23
−γ
ti e
= πh
Γ(γij )Pij ij .
ti
(B.7)
i<j
More generally,
2π h
n Z
Y
i=1
0
+∞
dti ℓi −
t e
ti i
Pn
i<j ti tj Pij


Z Y
n
n
n
Y
X
dγij
−γ
= πh
Γ(γij )Pij ij
δ
γij − ℓi  , (B.8)
2πi
i<j
i=1
j̸=i
and thus proves the Symanzik’s formula (3.10).
C
C.1
Flat-space limit
Review: flat-space limit of scalar correlators
In [37], it was shown that in the limit of γij → ∞, a scalar Mellin amplitude M (∆i , γij )
reproduces the flat-space amplitude A (pi ). In particular, in the case of massless scalar
scattering, the flat-space limit is given by
Z ∞
n
1 Pn
π h Y C∆i
Mn (∆i , γij ) ≈
dββ 2 ( i=1 ∆i −d)−1 e−β An (pi · pj = 2βγij ) .
(C.1)
2
Γ(∆i ) 0
i=1
– 33 –
For example, in ϕ3 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 A3 = g in the r.h.s. of (C.1) and get
d
Z ∞
3
3
1 P3
π h Y C∆i
π 2 Y C∆i
dββ 2 ( i=1 ∆i −d)−1 e−β = g
g
Γ
2
Γ(∆i ) 0
2
Γ(∆i )
i=1
P3
i=1 ∆i
2
i=1
−d
!
,
(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 ∆, the Mellin amplitude is [36]
4
∞
π h Y C∆i X
g2
MExch (∆i ) =
2 −∆
2
Γ(∆i )
4n! Γ(1 + ∆ − h + n) γ12 − ∆1 +∆
+n
2
n=0
i=1
∆1 + ∆ 2 + ∆ − d
∆1 + ∆ 2 − ∆
×Γ
1−
2
2
n
∆3 + ∆ 4 + ∆ − d
∆3 + ∆ 4 − ∆
×Γ
1−
. (C.3)
2
2
n
In the flat-space limit, γ12 → ∞,
MExch (∆i ) ≈
4
∞
1
π h Y C∆i g 2 X
2
Γ(∆i ) 4γ12
n!Γ(1 + ∆ − d/2 + n)
n=0
i=1
n Z ∞
1
∂
(∆1 +∆2 +∆−d)−1 −β1 12 (∆1 +∆2 −∆)−1
2
×
dβ1 β1
e t1
∂tn1 0
t1 =1
n Z ∞
1
1
∂
(∆
+∆
+∆−d)−1
(∆
+∆
−∆)−1
3
4
3
4
−β2 2
2
dβ2 β2
e t2
×
∂tn2 0
. (C.4)
t2 =1
Rescaling the integration variables β1,2 → β1,2 /t1,2 and using the following identity proven
in Appendix C.3 of [37], namely
∞
X
n=0
1
n!Γ(1 + ∆ − d/2 + n)
∂ n ∂ n − βt 1 − βt 2 −∆+ d2 −1 −∆+ d2 −1
e 1 2 t1
t2
∂tn1 ∂tn2
d
β12
=
t1 =t2 =1
−∆ −β1
e
δ(β1 − β2 ) , (C.5)
we get
MExch (∆i ) ≈
Z ∞
4
1 P4
g 2 −β
π h Y C∆i
e .
dββ 2 ( i=1 ∆i −d)−1
2
Γ(∆i ) 0
4βγ12
(C.6)
i=1
Note that it is consistent with (C.1) with AExch (pi ) = g 2 (2p1 ·p2 )−1 the flat-space amplitude
in the s-channel.
– 34 –
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
!Z
3
∞
h
Y
p
1
π
M1 M2 M3
M3v
=
DMi Ai
dββ 2 (γ1 +γ2 +γ3 −d)−1 e−β A3,A1 A2 A3 (iki → 2 βPi ) .
2
0
i=1
(C.7)
For the four-point gluon amplitude of the contact diagram, the Mellin amplitude (3.18b)
can also be expressed as
!Z
4
∞
1 P4
π h Y Mi Ai
M1 M2 M3 M4
MExch
=
D
dββ 2 ( i=0 ∆i −d)−1 e−β Acontact,A1 A2 A3 A4 . (C.8)
2
0
i=1
Now we look at the s-channel of the vector four-point Mellin amplitude (3.13). In the
limit of γ12 → ∞, it becomes
! ∞
4
on
o
h
Y
X Γ (d − 1) ( d − n)n 2 n
M1 M2 M3 M4
2π
a1 a2 b a3 a4 b
Mi Ai
2
MExch
≈ −g
f
f
D
X
·
X
12
34 .
2
4n!Γ d2 + n γ12
i=1
n=0
(C.9)
The sum over n can be implemented by using (C.5) with ∆ = d − 1. So,
2
n Z ∞
∞ ∞
X
X
d
Γ (d − 1) ( d2 − n)n
∂
1
d−2 −β1 2 −1
dβ1 β1 e t1
=
n!Γ(d/2 + n) ∂tn1 0
n!Γ d2 + n
t1 =1
n=0
n=0
n Z ∞
Z
∞
d
3d−4
∂
−1
×
=
dββ 2 −1 e−β . (C.10)
dβ2 β2d−2 e−β2 t22
n
∂t2 0
0
t2 =1
Therefore,
M1 M2 M3 M4
MExch
!Z
4
∞
h
Y
3d−4
π
IA1 A2 A3 A4
≈ − g 2 f a1 a2 b f a3 a4 b
DMi Ai
dββ 2 −1 e−β
2
4γ12
0
i=1
!
Z ∞
4
3d−4
π h Y Mi Ai
dββ 2 −1 e−β
=
D
2
0
i=1
p
i
1
→
, iki → 2 βPi .
× As−channel,A1 A2 A3 A4
k1 · k2
2βγ12
(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 γij → ∞,
!Z
Pn
n
∞
π h Y Mi Ai
i=1 ∆i −d −1 −β
M1 ···Mn
2
Mn
≈
D
dββ
e
2
0
i=1
(C.12)
p
i
1
P
× An,A1 A2 ···An
, iki → 2 βPi .
P 2 →
4β i<j γij
ki
i
– 35 –
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
Z ∞
(n−1)d−n
−1 −β
˜
2
dββ
Mn,A1 A2 ···An (P1 , P2 , · · · , Pn−1 , γij ) ≈
e
0
p
i
1
P
, iki → 2 βPi . (C.13)
× An,A1 A2 ···An
P 2 →
4β i<j γij
k
i
i
By plugging it in (4.7), we get
n+1
Y
πh
≈
2
M1 M2 ···Mn+1
Mn+1
×
Mi Ai
igf
D
an an+1 b
∞
n
oX
M
Xn(n+1)
m=0
i=1
X
Pn−1
r<s
×
!
n−1
Y
nrs =m r<s
a a ···a
b
AA11A22 ···An−1
n−1 M
(γrs )nrs
nrs !
Z
∞
dββ
4Γ
(n−1)d−n
−1
2
d
2
V3m,0,0
+ m γn(n+1)
e−β
0
i
p
1
P
, iki → 2 βPi
P 2 →
4β i<j (γij + nij )
i ki
. (C.14)
For the sum over nij , we can repeatedly use
X
P
i<j
(
1
P
=
nij !
i<j (γij + nij )
nij fixed i<j
P
γij )Pi<j nij
1
P
P
( i<j nij )!
i<j (γij + nij )
P
( i<j γij )Pi<j nij
1
P
P
≈
( i<j nij )!
(γ
i<j ij + nij ) − 1
P
( i<j γij − 1)Pi<j nij
1
P
P
=
( i<j nij )!
i<j γij − 1
P
P
( i<j γij − 1) i<j nij
1
P
P
≈
,
( i<j nij )!
i<j γij
Y (γij )nij
i<j
(C.15)
for each propagator term in (C.14). Therefore, with totally number of propagators Np , we
have
X
Pn−1
r<s
n−1
Y
nrs =m r<s
(γrs )nrs
nrs !
Y
1
i<j (γij + nij )
P
propagators
Pn−1
≈
r<s
γrs − Np
m!
m
1
Y
P
propagators
i<j
γij
. (C.16)
Here, the sum of γrs can be evaluated at the pole
n−1
X
i<j
(n − 2)(d − 1) −
γij =
2
– 36 –
Pn−1
i=1
δi
−m .
(C.17)
Thus, (C.14) becomes
!
n+1
∞
n
o
X
1
π h Y Mi Ai
1
M
≈
D
igf an an+1 b Xn(n+1)
d
2
4γn(n+1)
+
m
m!Γ
2
m=0
i=1
m Z ∞
m Z ∞
(n−1)d−n
d
∂
∂
−1
−1
dβ1 β1d−2 e−β1 t12
dβ2 β2 2
×
e−β2
m
∂tm
∂t
0
0
1
2
t1 =1
δ
(n−2)(d−1)−Pn−1
p
i=1 i −N −1
i
1
p
a1 a2 ···an−1 b
2
P
, iki → 2 β2 Pi t2
.
AA1 A2 ···An−1 M P 2 →
4β2 i<j γij
ki
t2 =1
M1 M2 ···Mn+1
Mn+1
i
(C.18)
Rescaling β1 → β1 /t1 , β2 → β2 /t2 in (C.18) and using (C.5) with ∆ = d − 1, we finally
arrive at the formula (C.12) for the (n + 1)-point Mellin amplitude Mn+1 .
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