Mellin Amplitudes: the Scattering Amplitudes of CFTmember.ipmu.jp/.../2015/26-06-2015-Joao-Penedones.pdf · Introduction Best language: Mellin amplitudes CFT correlation functions

Mellin Amplitudes:the Scattering Amplitudes of CFT

Strings 2015, Bengaluru, India, June 26th, 2015

João PenedonesUniversidade do Porto & CERN

Collaborators: Costa, Drummond, Fitzpatrick, Gonçalves, Kaplan, Raju, Trevisani, van Rees

Outline

• Introduction

• CFT

• Definition of Mellin amplitudes

• OPE & Factorization

• AdS/CFT

• Witten diagrams

• Flat space limit of AdS

• Bulk locality

• Problems for the future

Introduction

Best language: Mellin amplitudes

[Gary, Giddings, JP ‘09]CFT correlation functions can be thought of as AdS scattering amplitudes.

CFT

d

AdSd+1Idea: explore this analogy to learn about CFT and AdS/CFT

CFT

1 Introduction

Scattering amplitudes are transition amplitudes between states that describe non-interacting

and uncorrelated particles in the infinite past (in states) and states that describe non-

interacting and uncorrelated particles in the infinite future (out states). This definition

makes sense in Minkowski spacetime, where particles become infinitely distant from each

other in the infinite past and future. Anti-de Sitter (AdS) spacetime has a timelike confor-

mal boundary and does not admit in and out states. Pictorically, one can say that particles

in AdS live in an box and interact forever. Thus, in AdS, we can not use the standard

definition of scattering amplitudes. However, we can create and anihilate particles in AdS

by changing the boundary conditions at the timelike boundary. By the AdS/CFT corre-

spondence [1, 2, 3], the transition amplitudes between this type of states are equal to the

correlation functions of the dual conformal field theory (CFT). This suggests that we should

interpret the CFT correlation functions as AdS scattering amplitudes [4, 5, 6, 7]. In this

paper, we support this view using a representation of the conformal correlation functions

that makes their scattering amplitude nature more transparent.

We shall use the Mellin representation recently proposed by Mack in [8, 9].1

The

Euclidean correlator of primary scalar operators

A(xi) = �O1(x1) . . .On(xn)� , (1)

can be written as

A(xi) =N

(2πi)n(n−3)/2

�dδij M(δij)

n�

i<j

Γ(δij)�x2ij

�−δij(2)

where the integration contour runs parallel to the imaginary axis with Re δij > 0. Moreover,

the integration variables are constrained by

n�

j �=i

δij = ∆i , (3)

so that the integrand is conformally covariant with scaling dimension ∆i at the point xi.

This gives n(n − 3)/2 independent integration variables. We give the precise definition of

the integration measure in appendix A. Notice that n(n − 3)/2 is also the number of inde-

pendent conformal invariant cross-ratios that one can make using n points and the number

of independent Mandelstam invariants of a n-particle scattering process. The normalization

1The Mellin representation was used before, for example in [10, 11, 12], but its analogy with scatteringamplitudes was not emphasized.

1

Mellin Amplitudes [Mack ‘09]

Correlation function of scalar primary operators

A(xi) =

�[dγ]M(γij)

n�

i<j

Γ(γij)�x2ij

�−γij

1 Introduction
















The


A(xi) = �O1(x1) . . .On(xn)� , (1)

can be written as

A(xi) =N

(2πi)n(n−3)/2

�dδij M(δij)

n�

i<j

Γ(δij)�x2ij

�−δij(2)



n�

j �=i

δij = ∆i , (3)







1



A(xi) =

�[dγ]M(γij)

n�

i<j

Γ(γij)�x2ij

�−γij

1 Introduction
















The


A(xi) = �O1(x1) . . .On(xn)� , (1)

can be written as

A(xi) =N

(2πi)n(n−3)/2

�dδij M(δij)

n�

i<j

Γ(δij)�x2ij

�−δij(2)



n�

j �=i

δij = ∆i , (3)







1



M(γij)

A(xi) =

�[dγ]M(γij)

n�

i<j

Γ(γij)�x2ij

�−γij

1 Introduction
















The


A(xi) = �O1(x1) . . .On(xn)� , (1)

can be written as

A(xi) =N

(2πi)n(n−3)/2

�dδij M(δij)

n�

i<j

Γ(δij)�x2ij

�−δij(2)



n�

j �=i

δij = ∆i , (3)







1



M(γij)

Constraints:n�

j=1

γij = 0 , γii = −∆i = −dim[Oi] , γij = γji

A(xi) =

�[dγ]M(γij)

n�

i<j

Γ(γij)�x2ij

�−γij

1 Introduction
















The


A(xi) = �O1(x1) . . .On(xn)� , (1)

can be written as

A(xi) =N

(2πi)n(n−3)/2

�dδij M(δij)

n�

i<j

Γ(δij)�x2ij

�−δij(2)



n�

j �=i

δij = ∆i , (3)







1



n(n− 3)

2# integration variables = # ind. cross-ratios =

M(γij)

Constraints:n�

j=1

γij = 0 , γii = −∆i = −dim[Oi] , γij = γji

Example: n = 4 ∆i = ∆

A(xi) =1

(x213x

224)

∆

� i∞

−i∞

dγ12dγ14(2πi)2

�x212x

234

x213x

224

�−γ12�x214x

223

x213x

224

�−γ14

×

×M(γ12, γ14)Γ2(γ12)Γ

2(γ14)Γ2(∆− γ12 − γ14)


A(xi) =1

(x213x

224)

∆

� i∞

−i∞

dγ12dγ14(2πi)2

�x212x

234

x213x

224

�−γ12�x214x

223

x213x

224

�−γ14

×

×M(γ12, γ14)Γ2(γ12)Γ

2(γ14)Γ2(∆− γ12 − γ14)

γ12

0−1−2−3−4

∆− γ14


A(xi) =1

(x213x

224)

∆

� i∞

−i∞

dγ12dγ14(2πi)2

�x212x

234

x213x

224

�−γ12�x214x

223

x213x

224

�−γ14

×

×M(γ12, γ14)Γ2(γ12)Γ

2(γ14)Γ2(∆− γ12 − γ14)

γ12

0−1−2−3−4

∆− γ14

M(γ12, γ13, γ14)

γ12 + γ13 + γ14 = ∆

Crossing symmetry

Operator Product ExpansionO1(x1)O2(x2) =

�

k

C12k

(x212)

12 (∆1+∆2−∆k)

[Ok(x2) + descendants]


�

k

C12k

(x212)

12 (∆1+∆2−∆k)


�O1(x1)O2(x2) . . . � =�

dγ12�x212

�−γ12 . . .M(γ12, . . . )


�

k

C12k

(x212)

12 (∆1+∆2−∆k)


has simple poles at

with residue

⇒

∼ C12k�OkO3 . . .On�

M

descendants

γ12 =∆1 +∆2 −∆k − 2m

2, m = 0, 1, 2, . . .� ��

�O1(x1)O2(x2) . . . � =�

dγ12�x212

�−γ12 . . .M(γ12, . . . )


�

k

C12k

(x212)

12 (∆1+∆2−∆k)


Discrete spectrum of scaling dimensions implies that the Mellin amplitude is meromorphic.

has simple poles at

with residue

⇒

∼ C12k�OkO3 . . .On�

M

descendants

γ12 =∆1 +∆2 −∆k − 2m

2, m = 0, 1, 2, . . .� ��

�O1(x1)O2(x2) . . . � =�

dγ12�x212

�−γ12 . . .M(γ12, . . . )

Introduce such that and

. Then automatically solves the constraints.

OPE Factorization[Mack ‘09]

pi γij = pi · pj�n

i=1 pi = 0 γij

⇒(p2i = −∆i)

Introduce such that and

. Then automatically solves the constraints.

OPE Factorization[Mack ‘09]

pi γij = pi · pj�n

i=1 pi = 0 γij

⇒(p2i = −∆i)

M

1 n

k k + 1

∼ 1

(p1 + · · ·+ pk)2 +∆

1

k

n

k + 1

×ML MR∆ ∆

The OPE implies poles and factorization of residues.

[Fitzpatrick, Kaplan, JP, Raju, van Rees ’11][Fitzpatrick, Kaplan ‘12]

[Gonçalves, JP, Trevisani, ’14]

� �� ki,j=1 γij

�O1 . . .OkO∆� �O∆Ok+1 . . .On��O1 . . .On�

OPE Factorization⇒

[Gonçalves, JP, Trevisani, ’14]determined from and Qm ML MR

�O∆,lOk+1 . . .On� → MR�O1 . . .OkO∆,l� → ML

Mn ≈ Qm(γij)

(p1 + · · ·+ pk)2 +∆− l + 2m� �� m = 0, 1, 2, . . .� ��

twistdescendants

OPE Factorization⇒

[Gonçalves, JP, Trevisani, ’14]determined from and Qm ML MR

�O∆,lOk+1 . . .On� → MR�O1 . . .OkO∆,l� → ML

Mn ≈ Qm(γij)

(p1 + · · ·+ pk)2 +∆− l + 2m� �� m = 0, 1, 2, . . .� ��

twistdescendants

For and :n = 4 k = 2

O1

O2

O3

O4

∆, l

Qm(γij) = C12OC34OPl,m(γ13)

Mack polynomialKinematical polynomial of degree l

[Mack ‘09]

Large N CFT

In planar correlators of large N CFT

O1

O2

O3

O4

∆, l

∆ = ∆1 +∆2 + l + 2n+O

�1

N2

�

Large N CFT

In planar correlators of large N CFT

Only single-trace operators give rise to poles in planar Mellin amplitudes.

Γ(γ12) → OPE contributions of O1∂ . . . ∂O2

O1

O2

O3

O4

∆, l

∆ = ∆1 +∆2 + l + 2n+O

�1

N2

�

AdS/CFT

Witten diagramsBulk contact interaction

1

2n � 1

n

...

g

Figure 1: Witten diagram for a tree level n-point contact interaction in AdS.

One can also describe tensor fields in AdS using this language. A tensor field in AdS can

be represented by a transverse tensor field in Md+2,

XAiTA1...Al(X) = 0 . (17)

Covariant derivatives in AdS can be easily obtained from simple partial derivatives in the

flat embedding space. The rule is to take partial derivatives of transverse tensors and then

project into the tangent space of AdS using the projector

UAB = δBA +

XAXB

R2. (18)

For example

∇A3∇A2TA1(X) = UB3A3

UB2A2

UB1A1

∂B3

�UC2B2UC1B1

∂C2TC1(X)�. (19)

2.1 Contact interaction

Let us start by considering the simple Witten diagram in figure 1,

A(Pi) = g

�

AdS

dXn�

i=1

GB∂(X,Pi) , (20)

where g is a coupling constant. Using the representation (14), we obtain the following

expression for the n-point function

A(Pi) = gRn(1−d)/2+d+1

�n�

i=1

C∆i

�D∆1...∆n(Pi) , (21)

5

gφ1φ2 . . .φn

M(γij) = g

[JP ’10]

Witten diagramsBulk contact interaction

1

2n � 1

n

...

g




XAiTA1...Al(X) = 0 . (17)




UAB = δBA +

XAXB

R2. (18)

For example


UB2A2

UB1A1

∂B3

�UC2B2UC1B1

∂C2TC1(X)�. (19)



A(Pi) = g

�

AdS

dXn�

i=1

GB∂(X,Pi) , (20)



A(Pi) = gRn(1−d)/2+d+1

�n�

i=1

C∆i

�D∆1...∆n(Pi) , (21)

5

gφ1φ2 . . .φn

M(γij) = g

Contact diagrams in AdS give polynomial Mellin amplitudes

M(γij) = gPolynomial(γij)

g∇ . . .∇φ1 . . .∇ . . .∇φn

degree =1

2(# derivatives)

[JP ’10]

Witten diagramsFeynman rules for scalar exchanges

1

2

3

4

5

∆ m ∆ m

[Fitzpatrick, Kaplan, JP, Raju, van Rees ’11] [Paulos ’11]


1

2

3

4

5p1

p2

p3

p4

p5

p4+p5p1 + p2

∆ m ∆ m



1

2

3

4

5p1

p2

p3

p4

p5

p4+p5p1 + p2

∆ m ∆ m


M =∞�

m,m=0

V

�∆1

0

∆2

0

∆

m

�Sm(∆)

(p1 + p2)2 +∆+ 2mV

�∆

m

∆

m

∆3

0

�Sm(∆)

(p4 + p5)2 + ∆+ 2mV

�∆

m

∆4

0

∆5

0

�

Flat space limit of AdS

Anti-de SitterMinkowski

R → ∞Ki

K2i = 0 M2

i R2 = ∆i(∆i − d)

[Susskind ’99] [Polchinski ’99] [Gary, Giddings, JP ‘09] [Okuda, JP ’10]

[JP ’10] [Fitzpatrick, Kaplan ’12]

Flat space limit of AdS

Anti-de SitterMinkowski

R → ∞Ki

K2i = 0 M2

i R2 = ∆i(∆i − d)

Mellin amplitudeScattering amplitude

T (Ki) = limR→∞

N� i∞

−i∞

dα

2πieαα

d−�

∆i2 M

�γij =

R2Ki ·Kj

2α

�

[Susskind ’99] [Polchinski ’99] [Gary, Giddings, JP ‘09] [Okuda, JP ’10]

[JP ’10] [Fitzpatrick, Kaplan ’12]

Bulk Locality from CFTChapter 4. Eikonal Approximation in AdS/CFT

1 2

3 4

j,!

Figure 4.6: Witten diagram representing the T–channel exchange of an AdS particle with spinj and dimension !.

4.7 T–channel Decomposition

We have found that the eikonal approximation in AdS determines the small z and z behavior

of the reduced Lorentzian amplitude A. In the previous sections we have explored this result

using the S–channel partial wave expansion. We shall now study the T–channel partial wave

decomposition of the tree–level diagram in figure 4.6. The corresponding Euclidean amplitude

A1 can be expanded in T–channel partial waves,

A1 =!

µh,h Th,h . (4.23)

On the other hand, from the term of order g2 in (4.14), we have

A!1 ! i g2 22j!3 N!1N!2

"

"p2#!1

"

"p2#!2

$

M

dy dy e!2p·y!2p·y

|y|d!2!1+1!j |y|d!2!2+1!j""

%

y

|y| ,y

|y|

&

,

where we recall the expressions zz ! p2p2 and z + z ! 2p · p relating the cross ratios to the

points p and p in the past Milne wedge "M. Performing the radial integrals over |y| and |y|one obtains

A!1 ! i g2 K (zz)(1!j)/2

$

Hd!1

dw dw"" (w, w)

("2e ·w)2!1!1+j ("2e · w)2!2!1+j, (4.24)

with the constant K given by

K = 22j!3 N!1N!2 #(2!1 " 1 + j)#(2!2 " 1 + j) (4.25)

and e, e # Hd!1 defined by

e = " p

|p| , e = " p

|p| ,

so that

"2e · e =

'

z

z+

'

z

z.

84

Chapter 4. Eikonal Approximation in AdS/CFT

1 2

3 4

j,!








A1 =!

µh,h Th,h . (4.23)


A!1 ! i g2 22j!3 N!1N!2

"

"p2#!1

"

"p2#!2

$

M

dy dy e!2p·y!2p·y

|y|d!2!1+1!j |y|d!2!2+1!j""

%

y

|y| ,y

|y|

&

,



A!1 ! i g2 K (zz)(1!j)/2

$

Hd!1

dw dw"" (w, w)

("2e ·w)2!1!1+j ("2e · w)2!2!1+j, (4.24)


K = 22j!3 N!1N!2 #(2!1 " 1 + j)#(2!2 " 1 + j) (4.25)


e = " p

|p| , e = " p

|p| ,

so that

"2e · e =

'

z

z+

'

z

z.

84

+

Poles Polynomial

�O1 . . .O4�planar =

• Large N + finite number of low dimension single-trace operators

⇒∆gap → ∞ Finite # of exchange diagrams

[Heemskerk, JP, Polchinski, Sully ’09][Heemskerk, Sully ’10] [JP ’10]


1 2

3 4

j,!








A1 =!

µh,h Th,h . (4.23)


A!1 ! i g2 22j!3 N!1N!2

"

"p2#!1

"

"p2#!2

$

M

dy dy e!2p·y!2p·y

|y|d!2!1+1!j |y|d!2!2+1!j""

%

y

|y| ,y

|y|

&

,



A!1 ! i g2 K (zz)(1!j)/2

$

Hd!1

dw dw"" (w, w)

("2e ·w)2!1!1+j ("2e · w)2!2!1+j, (4.24)


K = 22j!3 N!1N!2 #(2!1 " 1 + j)#(2!2 " 1 + j) (4.25)


e = " p

|p| , e = " p

|p| ,

so that

"2e · e =

'

z

z+

'

z

z.

84


1 2

3 4

j,!








A1 =!

µh,h Th,h . (4.23)


A!1 ! i g2 22j!3 N!1N!2

"

"p2#!1

"

"p2#!2

$

M

dy dy e!2p·y!2p·y

|y|d!2!1+1!j |y|d!2!2+1!j""

%

y

|y| ,y

|y|

&

,



A!1 ! i g2 K (zz)(1!j)/2

$

Hd!1

dw dw"" (w, w)

("2e ·w)2!1!1+j ("2e · w)2!2!1+j, (4.24)


K = 22j!3 N!1N!2 #(2!1 " 1 + j)#(2!2 " 1 + j) (4.25)


e = " p

|p| , e = " p

|p| ,

so that

"2e · e =

'

z

z+

'

z

z.

84

+

Poles Polynomial


•Regge limit: [Cornalba ’07] [Costa, Gonçalves, JP ’12][Maldacena, Shenker, Stanford ’15]s → ∞ , t fixed

M(s, t) ∼ sj0 j0 ≤ 2





1 2

3 4

j,!








A1 =!

µh,h Th,h . (4.23)


A!1 ! i g2 22j!3 N!1N!2

"

"p2#!1

"

"p2#!2

$

M

dy dy e!2p·y!2p·y

|y|d!2!1+1!j |y|d!2!2+1!j""

%

y

|y| ,y

|y|

&

,



A!1 ! i g2 K (zz)(1!j)/2

$

Hd!1

dw dw"" (w, w)

("2e ·w)2!1!1+j ("2e · w)2!2!1+j, (4.24)


K = 22j!3 N!1N!2 #(2!1 " 1 + j)#(2!2 " 1 + j) (4.25)


e = " p

|p| , e = " p

|p| ,

so that

"2e · e =

'

z

z+

'

z

z.

84


1 2

3 4

j,!








A1 =!

µh,h Th,h . (4.23)


A!1 ! i g2 22j!3 N!1N!2

"

"p2#!1

"

"p2#!2

$

M

dy dy e!2p·y!2p·y

|y|d!2!1+1!j |y|d!2!2+1!j""

%

y

|y| ,y

|y|

&

,



A!1 ! i g2 K (zz)(1!j)/2

$

Hd!1

dw dw"" (w, w)

("2e ·w)2!1!1+j ("2e · w)2!2!1+j, (4.24)


K = 22j!3 N!1N!2 #(2!1 " 1 + j)#(2!2 " 1 + j) (4.25)


e = " p

|p| , e = " p

|p| ,

so that

"2e · e =

'

z

z+

'

z

z.

84

+

Poles Polynomial



M(s, t) ∼ sj0 j0 ≤ 2 ⇒ Finite # of contact diagrams





1 2

3 4

j,!








A1 =!

µh,h Th,h . (4.23)


A!1 ! i g2 22j!3 N!1N!2

"

"p2#!1

"

"p2#!2

$

M

dy dy e!2p·y!2p·y

|y|d!2!1+1!j |y|d!2!2+1!j""

%

y

|y| ,y

|y|

&

,



A!1 ! i g2 K (zz)(1!j)/2

$

Hd!1

dw dw"" (w, w)

("2e ·w)2!1!1+j ("2e · w)2!2!1+j, (4.24)


K = 22j!3 N!1N!2 #(2!1 " 1 + j)#(2!2 " 1 + j) (4.25)


e = " p

|p| , e = " p

|p| ,

so that

"2e · e =

'

z

z+

'

z

z.

84


1 2

3 4

j,!








A1 =!

µh,h Th,h . (4.23)


A!1 ! i g2 22j!3 N!1N!2

"

"p2#!1

"

"p2#!2

$

M

dy dy e!2p·y!2p·y

|y|d!2!1+1!j |y|d!2!2+1!j""

%

y

|y| ,y

|y|

&

,



A!1 ! i g2 K (zz)(1!j)/2

$

Hd!1

dw dw"" (w, w)

("2e ·w)2!1!1+j ("2e · w)2!2!1+j, (4.24)


K = 22j!3 N!1N!2 #(2!1 " 1 + j)#(2!2 " 1 + j) (4.25)


e = " p

|p| , e = " p

|p| ,

so that

"2e · e =

'

z

z+

'

z

z.

84

+

Poles Polynomial



M(s, t) ∼ sj0 j0 ≤ 2 ⇒ Finite # of contact diagrams




[Camanho, Edelstein, Maldacena, Zhiboedov ’14]

Generalization of causality constraints for �TµνTαβTρσ�

Problems for the future• Extend factorization formulas to general spin

• AdS tree-level n-point Mellin amplitudes a la

• When is Conformal Regge Theory valid?

• Simplify the OPE in large N CFTs

• Understand scaling of higher derivative corrections in AdS with

• Mellin amplitude of Virasoro blocks (minimal models)

• Study boundary correlators of QFT in AdS from BCFT correlators (UV) to scattering amplitudes (IR)

[Alday, Zhiboedov ’15]

[Cachazo, He, Yuan ’14]

∆gap � 1 [Camanho, Edelstein, Maldacena, Zhiboedov ’14]

Problems for the future• Extend factorization formulas to general spin

• AdS tree-level n-point Mellin amplitudes a la

• When is Conformal Regge Theory valid?

• Simplify the OPE in large N CFTs

• Understand scaling of higher derivative corrections in AdS with

• Mellin amplitude of Virasoro blocks (minimal models)

• Study boundary correlators of QFT in AdS from BCFT correlators (UV) to scattering amplitudes (IR)

[Alday, Zhiboedov ’15]

[Cachazo, He, Yuan ’14]

∆gap � 1 [Camanho, Edelstein, Maldacena, Zhiboedov ’14]

Thank you!

Extra slides

x1

x2

x3

x4

x+x−“Maldelstam invariants”

s = −(p1 + p3)2 = ∆1 +∆3 − 2γ13

t = −(p1 + p2)2 = ∆1 +∆2 − 2γ12

Regge limit: s → ∞ , t fixed

Planar 4-pt Mellin amplitude determined from its poles up to a finite number of coefficients (subtractions).

Regge limit of 4-pt function

x1

x2

x3

x4


s = −(p1 + p3)2 = ∆1 +∆3 − 2γ13

t = −(p1 + p2)2 = ∆1 +∆2 − 2γ12




[Costa, Gonçalves, JP ’12]

M(s, t) ∼ sj0

[Cornalba ’07]

J

2

4

6

∆− d

2d

20

j0

Unitary bound

Figure 1: Shape of the leading Regge trajectory with vacuum quantum numbers in a CFT.

is the analytic continuation of the dimension of the leading-twist operators of spin J .1

In this paper, we focus on the four-point function of the stress-tensor multiplet in N = 4Super Yang-Mills (SYM), which we review in section 2. This is an interesting laboratory

because the four-point function is known up to three loops in terms of (multiple) polylog-

arithms [8] and up to six loops (in the planar limit) as an integral representation [9]. The

anomalous dimension of leading-twist operators in SYM is also known up to four loops [10].

Furthermore, the relevant structure constants or OPE coefficients have been computed up

to three loops [11].

In the weak coupling limit of SYM, the leading Regge trajectory breaks up into three

branches as shown in figure 2. In particular, the horizontal branch that controls the Regge

limit and the diagonal branch that controls the Lorentzian OPE limit become almost in-

dependent. In particular, the perturbative expansion around each branch of the leading

Regge trajectory contains very different information. There is only a small set of consistency

conditions that must be satisfied at the meeting point at ∆ = 3 and J = 1 [12]. From the

four-point function point of view, this means that at each order in perturbation theory the

Lorentzian OPE limit and the Regge limit provide independent information that can be used

to constrain its general form.

At weak coupling, all leading-twist operators have twist equal to two plus a small anoma-

lous dimension. In section 3, we review how this leads to logarithms in the Lorentzian OPE

limit. In particular, we explain how one can predict the leading logarithms in the Lorentzian

OPE limit at higher loop orders using lower-loop information. In section 4, we perform a

similar analysis of the Regge limit in perturbation theory. In particular, we show that the

Regge limit of the four-point function up to three loops is compatible with the predictions

of Conformal Regge theory [7], and compute the pomeron residue at next-to-leading order.

Moreover, we predict the form of the leading logarithms in the Regge limit at higher loop

1Another possibility is that a trajectory of higher-twist operators has an analytic continuation with largerintercept j0 and therefore dominates in the Regge limit. We think this is unlikely but are not aware of anyproof.

3

x1

x2

x3

x4


s = −(p1 + p3)2 = ∆1 +∆3 − 2γ13

t = −(p1 + p2)2 = ∆1 +∆2 − 2γ12


[Maldacena, Shenker, Stanford ’15]

Planar limit j0 ≤ 2⇒



[Costa, Gonçalves, JP ’12]

M(s, t) ∼ sj0

[Cornalba ’07]

J

2

4

6

∆− d

2d

20

j0

Unitary bound

Figure 1: Shape of the leading Regge trajectory with vacuum quantum numbers in a CFT.

is the analytic continuation of the dimension of the leading-twist operators of spin J .1

In this paper, we focus on the four-point function of the stress-tensor multiplet in N = 4Super Yang-Mills (SYM), which we review in section 2. This is an interesting laboratory

because the four-point function is known up to three loops in terms of (multiple) polylog-

arithms [8] and up to six loops (in the planar limit) as an integral representation [9]. The

anomalous dimension of leading-twist operators in SYM is also known up to four loops [10].

Furthermore, the relevant structure constants or OPE coefficients have been computed up

to three loops [11].

In the weak coupling limit of SYM, the leading Regge trajectory breaks up into three

branches as shown in figure 2. In particular, the horizontal branch that controls the Regge

limit and the diagonal branch that controls the Lorentzian OPE limit become almost in-

dependent. In particular, the perturbative expansion around each branch of the leading

Regge trajectory contains very different information. There is only a small set of consistency

conditions that must be satisfied at the meeting point at ∆ = 3 and J = 1 [12]. From the

four-point function point of view, this means that at each order in perturbation theory the

Lorentzian OPE limit and the Regge limit provide independent information that can be used

to constrain its general form.

At weak coupling, all leading-twist operators have twist equal to two plus a small anoma-

lous dimension. In section 3, we review how this leads to logarithms in the Lorentzian OPE

limit. In particular, we explain how one can predict the leading logarithms in the Lorentzian

OPE limit at higher loop orders using lower-loop information. In section 4, we perform a

similar analysis of the Regge limit in perturbation theory. In particular, we show that the

Regge limit of the four-point function up to three loops is compatible with the predictions

of Conformal Regge theory [7], and compute the pomeron residue at next-to-leading order.

Moreover, we predict the form of the leading logarithms in the Regge limit at higher loop

1Another possibility is that a trajectory of higher-twist operators has an analytic continuation with largerintercept j0 and therefore dominates in the Regge limit. We think this is unlikely but are not aware of anyproof.

3

Planar OPE?

2pt and 3pt functionsof single-trace operators

n-pt functionsof single-trace operators?

Planar OPE?



OPE

Planar OPE?



String Theory?

OPE

Planar OPE?



String Theory?

OPE

s�

sM(s, t) =

�ds�

2πi

M(s�, t)

s� − s

M(s, t) ∼ sj0 j0 ≤ 2

M(s, t) =�

s∗

Ress=s∗M(s, t)

s− s∗+

contribution

from s = ∞

For the Mellin amplitude is not unique.

Mellin Redundancy

=

�n(n−3)

2 , n ≤ d+ 2

nd− (d+2)(d+1)2 , n ≥ d+ 2

# independent cross ratios

n > d+ 2

det1≤i,j≤d+3

Pi · Pj = 0 Pi ∈ Rd+2

We can add to the Mellin amplitude the Mellin transform of this determinant times any function.

External Spin

�OA(P )O(P1) . . .O(Pn)� =n�

a=1

PAa

�[dγ]Ma(γij)

n�

i<j

Γ(γij)

(−2Pi · Pj)γij

n�

i=1

Γ(γi + δai )

(−2Pi · P )γi+δai

γi = −n�

j=1

γij , γij = γji , γii = −∆i ,n�

i,j=1

γij = 1−∆

M

1 n

k k + 1

∼

[Gonçalves, JP, Trevisani, ’14]

1

k

n

k + 1

×∆ ∆Ma

L M bR

k�

a=1

n�

b>k

γab

(p1 + · · ·+ pk)2 +∆− 1

Flat Space Limit of AdS

1

2n � 1

n

...

g




XAiTA1...Al(X) = 0 . (17)




UAB = δBA +

XAXB

R2. (18)

For example


UB2A2

UB1A1

∂B3

�UC2B2UC1B1

∂C2TC1(X)�. (19)



A(Pi) = g

�

AdS

dXn�

i=1

GB∂(X,Pi) , (20)



A(Pi) = gRn(1−d)/2+d+1

�n�

i=1

C∆i

�D∆1...∆n(Pi) , (21)

5

g∇ . . .∇φ1∇ . . .∇φ2 . . .∇ . . .∇φn

Evidence for

1) Works for an infinite set of interactions

T = lim

�. . .M

2) Derived from wave-packet scattering localized in small region of AdS [Fitzpatrick, Kaplan ’12]

[Gary, Giddings, JP ‘09] [Okuda, JP ’10]

gg�

Figure 3: A tree level scalar exchange in AdS contributing to a n-point correlation function.

This shows that the only poles of the Mellin amplitude are at

�

i∈L

�

j∈R

δij = ∆+ 2m , m = 0, 1, 2, . . . . (48)

Let us introduce ”momentum” ki associated with operator Oi, such that −k2i = ∆i and

�i ki = 0. Then, if we write δij = ki · kj as in (4), the pole condition reads

�

i∈L

�

j∈R

δij =�

i∈L

�

j∈R

ki · kj = −��

i∈L

ki�2

= ∆+ 2m , m = 0, 1, 2, . . . . (49)

This has the suggestive interpretation of the total exchanged ”momentum going on-shell”.

Finally, let us now return to the graviton exchange process discussed in the introduction.

With our conventions, the Mellin amplitude associated with graviton exchange between

minimally coupled massless scalars in AdS5 (∆i = d = 4), is given by

M(sij) = −32πG5R−3

5

�6γ2

13 + 2

s13 − 2+

8γ213

s13 − 4+

γ213 − 1

s13 − 6− 15

4s13 +

55

2

�, (50)

where G5 is the Newton’s constant in AdS5 and γ13 = (s12− s14)/2. The large sij limit gives

M(sij) ≈ 96πG5R−3 s12s14

s13, (51)

in agreement with the result of formula (9) using the scattering amplitude

T (Sij) = 8πG5S12S14

S13, (52)

for graviton exchange between minimally coupled massless scalars in flat space [23, 24].

11

4

21

3

gg�

��

Figure 4: One-loop Witten diagram contributing to the 4-point correlation function.

2.3 One-loop Witten diagram

It is important to test our main formula (9) beyond tree level diagrams. To this end, we

shall study the 1-loop diagram of figure 4. The associated Mellin amplitude is computed in

appendix D. The result reads

M(sij) =g2R6−2d

Γ��

i ∆i

2 − h�Γ�∆1+∆3−s13

2

�Γ�∆2+∆4−s13

2

�

i∞�

−i∞

dc

2πil(c)l(−c)q(c) , (53)

where l(c) is given by the same expression (39) as in the tree level exchange and

q(c) =Γ(c)Γ(−c)

8πhΓ(h)Γ(h+ c)Γ(h− c)

i∞�

−i∞

dc1dc2(2πi)2

Θ(c, c1, c2)

((∆− h)2 − c21) ((∆� − h)2 − c22)

, (54)

with

Θ(c1, c2, c3) =

�{σi=±} Γ

�h+σ1c1+σ2c2+σ3c3

2

��3

i=1 Γ(ci)Γ(−ci). (55)

Here,�

{σi=±} denotes the product over the 23 = 8 possible values of (σ1, σ2, σ3). This

one-loop Mellin amplitude is rather long but the fact that it is possible to write it down in

such a closed form is remarkable. Our goal with this example is simply to understand the

singularity structure and the flat space limit of one-loop Mellin amplitudes.

The singularities of M(sij) are simple poles as for the tree level diagrams. This is a

consequence of the discrete spectrum of a field theory in AdS. As before all poles are due

to pinching of the integration contour between two colliding poles of the integrand. Let us

start by finding the singularity structure of q(c). Firstly, we consider the integral over c2

with fixed c1. The integrand has poles at

±c2 = ∆� − h , ±c2 = h± c± c1 + 2m , m = 0, 1, 2, . . . (56)

12

3) Verified in several non-trivial examples

4

21

3

gg�

��

Figure 4: One-loop Witten diagram contributing to the 4-point correlation function.

2.3 One-loop Witten diagram

It is important to test our main formula (9) beyond tree level diagrams. To this end, we

shall study the 1-loop diagram of figure 4. The associated Mellin amplitude is computed in

appendix D. The result reads

M(sij) =g2R6−2d

Γ��

i ∆i

2 − h�Γ�∆1+∆3−s13

2

�Γ�∆2+∆4−s13

2

�

i∞�

−i∞

dc

2πil(c)l(−c)q(c) , (53)

where l(c) is given by the same expression (39) as in the tree level exchange and

q(c) =Γ(c)Γ(−c)

8πhΓ(h)Γ(h+ c)Γ(h− c)

i∞�

−i∞

dc1dc2(2πi)2

Θ(c, c1, c2)

((∆− h)2 − c21) ((∆� − h)2 − c22)

, (54)

with

Θ(c1, c2, c3) =

�{σi=±} Γ

�h+σ1c1+σ2c2+σ3c3

2

��3

i=1 Γ(ci)Γ(−ci). (55)

Here,�

{σi=±} denotes the product over the 23 = 8 possible values of (σ1, σ2, σ3). This

one-loop Mellin amplitude is rather long but the fact that it is possible to write it down in

such a closed form is remarkable. Our goal with this example is simply to understand the

singularity structure and the flat space limit of one-loop Mellin amplitudes.

The singularities of M(sij) are simple poles as for the tree level diagrams. This is a

consequence of the discrete spectrum of a field theory in AdS. As before all poles are due

to pinching of the integration contour between two colliding poles of the integrand. Let us

start by finding the singularity structure of q(c). Firstly, we consider the integral over c2

with fixed c1. The integrand has poles at

±c2 = ∆� − h , ±c2 = h± c± c1 + 2m , m = 0, 1, 2, . . . (56)

12

UV and IR divergences

UV divergences are the same in AdS and in flat space.

IR divergences are absent in AdS.

The Mellin amplitudes can be thought as IR regulated scattering amplitudes.

Application: from SYM to IIB strings N = 4 SYM

g2YM = 4πgsg2YMN = λ = (R/�s)

4

Lagrangian densityO(x) = φ = Dilaton

�O(x1)O(x2)O(x3)O(x4)�4pt function 2 2 scattering

amplitude

type IIB strings

R → ∞

Mellin amplitude

limλ→∞

λ32M(g2YM,λ, sij =

√λαij) =

1

120π3�6s

∞�

0

dββ5e−βT10

�gs, �s, Sij =

2β

�2sαij

�

FSL of AdS for massive particlesAnti-de SitterMinkowski

R → ∞Ki

Mellin amplitudeScattering amplitude

M2i R

2 = ∆i(∆i − d)

T (Ki) = limR→∞

N M

�∆i = RMi, γij = R

MiMj +Ki ·Kj�l Ml

�

K2i = −M2

i

[work in progress]

Documents

Mellin Amplitudes: the Scattering Amplitudes of CFTmember.ipmu.jp/.../2015/26-06-2015-Joao-Penedones.pdf · Introduction Best language: Mellin amplitudes CFT correlation functions