Nonperturbative Mellin Amplitudes: Existence, Properties, … · 2020. 1. 31. · Nonperturbative Mellin Amplitudes: Existence, Properties, Applications A. Zhiboedov (CERN) SISSA,

Nonperturbative Mellin Amplitudes: Existence, Properties, Applications

A. Zhiboedov (CERN) SISSA, Trieste

work with J. Penedones and J. Silva

Introduction

1

lD�2Pl

ZdDx

p�g (R� 2⇤+ ...)

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AdS/CFT

1

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ZdDx

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AdS/CFT

nonperturbative

perturbative

1

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ZdDx

p�g (R� 2⇤+ ...)




AdS/CFT

nonperturbative

perturbative

coarse-grained holography: assume a UV completion and derive rules of consistent AdS EFTs (Hofman-Maldacena type bounds) (easy)

1

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ZdDx

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AdS/CFT

nonperturbative

perturbative


fine-grained holography: given a consistent gravitational EFT that satisfies all the known bounds construct a UV completion. (hard)

Any EFT! Only String Theory!

1

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ZdDx

p�g (R� 2⇤+ ...)




AdS/CFT

nonperturbative

perturbative


fine-grained holography: given a consistent gravitational EFT that satisfies all the known bounds construct a UV completion. (hard)

BootstrapIs it possible to use bootstrap to shed some light upon the matter?

Tentative CFT data

NOMAYBE

[Rattazzi, Rychkov, Tonni, Vichi ’08]

Maybe

No

BootstrapIs it possible to use bootstrap to shed some light upon the matter?

Tentative CFT data

NOMAYBE


Maybe

No

no surprising bounds for large N theories yet

Bootstrap

“one is never sure to have completely exploited all the axioms of field theory.”

Old wisdom (S-matrix bootstrap):

Is it possible to use bootstrap to shed some light upon the matter?

Tentative CFT data

NOMAYBE


Maybe

No

no surprising bounds for large N theories yet

Mellin Representation

Mellin Space Coordinate space

Bootstrap

[Mack]

PerturbativeNonperturbative

What goes into the Mellin representation?

?

[Penedones][Paulos][Fitzpatrick, Kaplan, Penedones, Raju, van Rees]

[Aharony, Alday, Bissi, Perlmutter]

[Gopakumar, Kaviraj, Sen, Sinha][Rastelli, Zhou][Caron-Huot]

…

[Rattazzi, Rychkov, Tonni, Vichi]

…[Dodelson, Ooguri]

[Yuan]

[Poland, Rychkov, Vichi][Simmons-Duffin][Hogervorst, van Rees][Mazac, Paulos]

(Holographic)


Mack’s Mellin Ansatz: CFT correlation functions admit the following representation

[Mack ’09]

hO1(x1) . . .On(xn)i =

Z[d�ij ]M(�ij)

nY

i<j

�(�ij)

|xi � xj |2�ij

nX

j=1

�ij = 0 , �ij = �ji , �ii = ��i .


Mack’s Mellin Ansatz: CFT correlation functions admit the following representation

[Mack ’09]

hO1(x1) . . .On(xn)i =

Z[d�ij ]M(�ij)

nY

i<j

�(�ij)

|xi � xj |2�ij

nX

j=1

�ij = 0 , �ij = �ji , �ii = ��i .

[Penedones ’10]

Outline

Mellin transform (in CFTs)

Existence

Properties

Applications

Review of Mellin transform

Mellin Transform

Mellin transformation is defined as follows[Mellin ’1896]

�(x) : Rn+ ! C

M [�](z) =

Z

Rn+

dx xz�1�(x)

M�1[F ](x) =1

(2⇡i)n

Z

a+iRn

dz x�zF (z)

What are natural classes of functions associated with it?

M-space

Space of functions holomorphic in a sectorial domain , which are polynomially bounded with powers in an open domain U.

Θ

MU⇥ : ⇥ 2 Rn, U 2 Rn, 0 2 ⇥ �(x) 2 MU

⇥

x = (r1ei✓1 , ..., rne

i✓n) ✓ 2 ⇥ r 2 Rn+

|�(x)| < C(a)|x�a|, x = (r, ✓) 2 S⇥, a 2 U

θU

W-space

Similarly, we define a space of functions holomorphic in a tube and decaying exponentially fast in the imaginary directionW⇥

U : F (z), z 2 U + iRn

|F (u+ iv)| K(u)e�sup✓2⇥✓·v u 2 U

Uθ

Mellin Inversion Theorems[Antipova ’07]

Theorem I: Given , the Mellin transform exists and . We also have .

Φ(x) ∈ MUΘ

M[Φ] ∈ WΘU M−1M[Φ] = Φ

Theorem II: Given , the inverse Mellin transform exists and . We also have .

F(z) ∈ WΘU

M−1[F] ∈ MUΘ MM−1[F] = F

θU

� < poly F < exp

Examples

e�u =

Z cu+i1

cu�i1

d�122⇡i

�(�12)u��12 , cu > 0, |arg[u]| < ⇡

2.

�(�12) ⇠ e�⇡2 |Im[�12]|

The Cahen-Mellin integral:

Examples

e�u =

Z cu+i1

cu�i1

d�122⇡i

�(�12)u��12 , cu > 0, |arg[u]| < ⇡

2.

�(�12) ⇠ e�⇡2 |Im[�12]|

The Cahen-Mellin integral:

�(2�12)�(↵� 2�12)

�(↵)⇠ e�2⇡|Im[�12]|

1

(1 +pu)↵

=

Z cu+i1

cu�i1

d�122⇡i

�(2�12)�(↵� 2�12)

�(↵)u��12 ,

0 < cu <↵

2, |arg[u]| < 2⇡

Mellin transform in CFTs

Are analyticity in a sectorial domain and polynomial boundedness properties of the physical correlators?

Four-point Function

Let us consider a four-point function of identical scalar primary operators

hO(x1)O(x2)O(x3)O(x4)i =F (u, v)

(x213x

224)

� , u =x212x

234

x213x

224

, v =x214x

223

x213x

224

F (u, v) = F (v, u) = v��F

✓u

v,1

v

◆

M̂(�12, �14) ⌘Z 1

0

dudv

uvu�12v�14F (u, v)

(crossing)

We can try to define the Mellin transform as follows

Region of Integration

(principal Euclidean sheet)

Mellin transform = Integral over the principal sheet

⇡⇡2

0�⇡ �

t

(1, 0)

(0, 1)

u

v

Analyticity Domain

Claim: Any CFT correlator is analytic in a sectorial domain .

F(u, v)ΘCFT = (arg[u], arg[v])

arg(v)

arg(u)2⇡�2⇡ 0

2⇡

�2⇡

(Reduced) Mellin Amplitude

M̂(�12, �14) = [�(�12)�(�13)�(�14)]2 M(�12, �14)

It is common in the field to define the following reduced Mellin amplitude

[�(�12)�(�14)�(�� 12 � �14)]2 ⇠ e�⇡(|Im[�12]|+|Im[�14]|+|Im[�12+�14]|)

= e�sup⇥CFT(arg[u]Im[�12]+arg[v]Im[�14])

The pre-factor correctly captures analytic properties of the correlator!

Polynomial Boundedness

We consider a subtracted correlator

Fsub(u, v) = F (u, v)� (1 + u�� + v��)

�X

⌧gap⌧<⌧sub

JmaxX

J=0

h⌧sub�⌧

2

i

X

m=0

C2OOO⌧,J

⇣u��+ ⌧

2+mg(m)⌧,J (v) + v��+ ⌧

2+mg(m)⌧,J (u) + v�

⌧2�mg(m)

⌧,J (u

v)⌘

crossing symmetric same analyticity properties improved behavior in the OPE limits

Dangerous Regions

⇡⇡2

0�⇡ �

t

(1, 0)

(0, 1)

u

v

Double light-cone limit u, v ! 0

[Alday, Eden, Korchemsky, Maldacena, Sokatchev ’10]

[Alday, AZ ’15]


Claim:

Saturated in minimal models and perturbative field theories

|Fsub(u, v)| C(cu, cv)1

|u|cu1

|v|cv , (u, v) 2 ⇥CFT , (cu, cv) 2 UUCFT

Mellin Amplitude

Fsub(u, v) =

Zd�122⇡i

d�142⇡i

M̂(�12, �14)u��12v��14

UCFT

UCFT :

3d Ising

As an example consider the spin operator in the 3d Ising model

σ⟨σσσσ⟩

F Ising3d (u, v) =

�1 + u�� + v��

�+

Z

C

d�122⇡i

d�142⇡i

M̂(�12, �14)u��12v��14 ,

C : �� 1

2< Re(�12),Re(�14),

Re(�12) + Re(�14) <1

2.

This representation converges in .ΘCFT

Given that CFT correlators in a general CFT admit the Mellin representation:

What are the properties of Mellin amplitudes?

Analyticity of

Polynomial boundedness of

Subtractions = Deformed contour

F(u, v)

F(u, v)

Properties of the CFT Mellin Amplitudes

Mellin vs OPEHow is the Mellin representation consistent with the OPE?

=

O(x1)

O(x2) O(x3)

O(x4)

O(x1)

O(x2) O(x3)

O(x4)

OiOi

X

i

X

i

Reproduce the OPE (unitarity)

Cancel the disconnected piece (Polyakov conditions)

F (u, v) =�1 + u�� + v��

�

+

Z

C

d�122⇡i

d�142⇡i

[�(�12)�(�14)�(�� 12 � �14)]2 M(�12, �14)u

��12v��14

OPELet us review in more detail the unitarity properties of Mellin amplitudes

t = 2�� 2�12,

s = 2(�12 + �14 ��) = �2�13

M(s, t) ' �C2

��⌧Q⌧,dJ,m(s)

t� (⌧ + 2m)+ ... , m = 0, 1, 2, ... ,

Q⌧,dJ,m(s) = K(�, J,m) Q⌧,d

J,m(s)

The residues are controlled by the so-called Mack polynomials

OPELet us review in more detail the unitarity properties of Mellin amplitudes

t = 2�� 2�12,

s = 2(�12 + �14 ��) = �2�13

M(s, t) ' �C2

��⌧Q⌧,dJ,m(s)

t� (⌧ + 2m)+ ... , m = 0, 1, 2, ... ,

Q⌧,dJ,m(s) = K(�, J,m) Q⌧,d

J,m(s)

The residues are controlled by the so-called Mack polynomials

Maximal Mellin Analyticity Conjecture: Mellin amplitudes have only poles located at twists of the physical operators (except the identity).

OPE

Mack polynomials have a few interesting propertiesQ⌧,d

J,m(s) = (�1)JQ⌧,dJ,m(�s� ⌧ � 2m)

QJ,0(s) =2J

�⌧2

�2J

(⌧ + J � 1)J3F2(�J, J + ⌧ � 1,�s

2;⌧

2,⌧

2; 1)

lims!1

Q�,⌧,dJ,m (s) = s

J +O(sJ�1)

Q⌧,dJ,m(s) = m

J2 (m+ ⌧)

J2 C

( d�22 )

J

⌧2 +m+ sp

m(m+ ⌧)

!.

Higher m’s can be computed recursively.

(high energy)

(flat space limit)

limJ,s!1

Q�,⌧,dJ,m (s) = 21+J�mJ2J+s+⌧+m�1e�2J ⇡

�( ⌧+2m+s2 )2

sm + ... (AdS effects)

Twist Spectrum of CFTsUnitarity

The twist spectrum in a CFT is very complex and contains an infinite number of accumulation points

limJ!1

⌧ (n)1,2 (J) = ⌧1 + ⌧2 + 2n

limJ!1

⌧ (m)

⌧ (n)1,2 (J0),⌧3

(J) = ⌧ (n)1,2 (J0) + ⌧3 + 2m

⌧ � d� 2

[Komargodski, AZ 12’][Fitzpatrick, Kaplan, Poland, Simmons-Duffin 12’][Callan, Gross] [Parisi]

τ1

τ2

τ1 + τ2

Bound on ReggeThe reduced Mellin amplitude M is polynomially bounded. The precise power follows from the bound on the Regge limit [Maldacena, Shenker, Stanford]

[Caron-Huot]

x+x�

x1x2

x4

x3

Conformal Regge Theory[Costa, Goncalves, Penedones ’12]

In Mellin space the Regge limit becomes

lims!1

M(s, t) 'Z 1

�1d⌫ �(⌫)!⌫,j(⌫)(t)

sj(⌫) + (�s)j(⌫)

sin (⇡j(⌫))+ ...

jfull(0) 1

jplanar(0) 2

lim|s|!1

|Mplanar(s, t)| < c|s|2+✏

lim|s|!1

|Mfull(s, t)| c|s|

Re[t] < d − 2

Extrapolated away from the imaginary axis.

Polyakov ConditionsIn the full nonperturbative correlator we have to make sure that we do not have double counting as we close the contour

F (u, v) =�1 + u�� + v��

�

+

Z

C

d�122⇡i

d�142⇡i

[�(�12)�(�14)�(�� 12 � �14)]2 M(�12, �14)u

��12v��14

These are the Polyakov conditions. Absent in the planar theory.

[cf. Gopakumar, Kaviraj, Sen, Sinha]

The subtlety is related to the accumulation points in the twist space.

Polyakov ConditionsAs we close the contour we get the divergent sum

−12

(CGFFτ=2Δ,J)

2Qτ,dJ,0(γ14)Γ2(γ14)Γ2 ( τ

2− γ14) Γ2 (Δ −

τ2 )

=4

Γ2(Δ)1J (J2(Δ−γ14)Γ2(γ14) + J2γ14Γ2(Δ − γ14) + . . . ) , J → ∞ , τ → 2Δ ,

thus avoiding the double counting problem.

Polyakov ConditionsAs we close the contour we get the divergent sum

−12

(CGFFτ=2Δ,J)

2Qτ,dJ,0(γ14)Γ2(γ14)Γ2 ( τ

2− γ14) Γ2 (Δ −

τ2 )

=4

Γ2(Δ)1J (J2(Δ−γ14)Γ2(γ14) + J2γ14Γ2(Δ − γ14) + . . . ) , J → ∞ , τ → 2Δ ,

thus avoiding the double counting problem.

A simple way to fix it is to consider

−u∂uF(u, v) = Δu−Δ + ∫C

dγ12

2πidγ14

2πiγ12M̂(γ12, γ14)u−γ12v−γ14 .

solves both convergence and double counting problems.

Polyakov ConditionsConvergence of the sum over spins leads to the following condition

τgap

2> Re[γ14] > Δ −

τgap

2

So that we get the following condition on the nonperturbative Mellin amplitude

lim|γ12|→0, arg[γ12]≠0

M(γ12, γ14) = 0,τgap

2> Re[γ14] > Δ −

τgap

2

In principle there are more conditions to explore.

Mellin amplitude reggeizes close to the accumulation points.

SummaryCFT connected correlators admit a deformed contour Mellin representationsFconn(u, v) ⌘ F (u, v)� (1 + u�� + v��)

=

Z Z

C

d�122⇡i

d�142⇡i

[�(�12)�(�14)�(�� 12 � �14)]2 M(�12, �14)u

��12v��14

Reduced Mellin amplitude M satisfies:

Maximal Mellin Analyticity (Unitarity)

Crossing Symmetry


Polyakov conditions

Applications

Minimal Models

All these general properties can be explicitly checked for minimal models. This is the only known example of Mellin amplitudes of fully non-perturbative correlators.

[Furlan, Petkova ’89][Lowe ’16][Alday, AZ ’15]

[Yuan ’18]

[Dotsenko, Fateev]

F 2d Ising�� (u, v) =

ppu+

pv + 1p

2 8puv

M̂(�12, �14) = �r

2

⇡�

✓2�12 �

1

4

◆�

✓2�14 �

1

4

◆�(�2�12 � 2�14)

Minimal Models [Furlan, Petkova ’89][Lowe ’16][Alday, AZ ’15]

[Yuan ’18]

[Dotsenko, Fateev]

M̂(�12, �14) = C0

Z[d⇠12]

Z[d⇠15]

Z[d⇠24]

Z[d⇠34]

Z[d⇠35]�(⇠12)�(⇠15)

�(�2↵↵+ + �12 � ⇠12)�(⇠24)�(⇠15 � ⇠24 � ⇠34 + 1)��2↵2

+ + 2↵↵+ + �12 � ⇠34 + 2�

�(⇠34)�(�2↵↵+ � ⇠12 � ⇠15 + ⇠34 � 1)�(�2↵↵+ + �14 � ⇠15 + ⇠24 + ⇠34 � 1)

��2↵2

+ + 2↵↵+ + �14 + ⇠12 + ⇠24 � ⇠35 + 1��2↵2

+ � ⇠15 + ⇠24 � ⇠35 � 1�

��⇠12 � ⇠34 � ⇠35 + 2↵↵+ � 2↵2

+ + 1��(⇠35)�(�⇠12 � ⇠24 + ⇠35 + 1)

��12 � �14 + ⇠15 � ⇠24 + ⇠35 + 4↵2

+ � 1�

��2↵2

+ + ⇠15��(�4↵↵+ � ⇠12 � ⇠15 + ⇠34 � 1)

�(��12 � �14 � ⇠24)

��2↵2

+ + ⇠35��4↵2

+ + 4↵↵+ + ⇠12 � ⇠34 � ⇠35 + 3� .

h�1,3O�1,3Oi

Lightray Operators

We used it compute energy correlators in N=4 SYM

hO†(p)E(~n1)E(~n2)O(p)i

[Belitsky, Henn, Hohenegger, Korchemsky, Sokatchev, Yan, AZ ’13-’19]

E ⇠ ANEC =

Zdy�T��

It is trivial to compute Wightman functions using Mellin amplitudes

hO1(x1) . . .On(xn)i =

Z[d�ij ]M(�ij)

nY

i<j

�(�ij)

(x2ij + i✏x0

ij)�ij

[Kologlu, Kravchuk, Simmons-Duffin, AZ]

Lightray Operators

We get the following expressions

Compare this with the OPE expression

hE(~n1)E(~n2)i =(p0)2

8⇡2

"X

i

p�i

4⇡4�(�i � 2)

�(�i�12 )3�( 3��i

2 )f4,4�i

(⇣) +1

4(2�(⇣)� �0(⇣))

#

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⇣ =1� cos�

2<latexit sha1_base64="WN3VzgxIu19GEHmO21XEK7vuyIA=">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</latexit>

f�1,�2

� (⇣) = ⇣��1��2+1

2 2F1

✓�� 1 +�1 ��2

2,�� 1��1 +�2

2,�+ 1� d

2, ⇣

◆

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• Sum is over Regge trajectories

[Kologlu, Kravchuk, Simmons-Duffin, AZ]

[Belitsky, Hohenegger, Korchemsky, Sokatchev, AZ]

AdS/EFT

Consider an effective field theory in AdS

S =

Zdd+1x

p�g(�@µ�@

µ�+ µ1�4 + µ2(@µ�@

µ�)2 + ...)

[Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi ’06]

Using dispersion relations we get

µ2 =2

⇡

Zds

s�(s)

s3> 0.

[Hartman, Jain, Kundu ’16]

Can we use similar arguments in Mellin space?

Positivity of Mack Polynomials

Mack polynomials share some positivity properties with Legendre polynomials

Q⌧,dJ,m(s) � 0, s � 0.

@ns Q

⌧,dJ,m(s)|s�0 � 0 .

We also observed evidence for more complicated EFThedron-like identities for Legendre polynomials.

[Arkani-Hamed, Huang, to appear]

Positive geometry for conformal bootstrap? [Arkani-Hamed, Huang, Shao ’18][Sen, Sinha, Zahed '19]

Dispersion Relations

Therefore, we can run an identical dispersion relation argument

µAdS2 =

1

2⇡i

Idt0

(t0)3M(0, t0) � 0

Similarly, higher order couplings receive the contribution suppressed by an extra power of the mass of the particle that we integrated out

μAdSn (∂μϕ∂μϕ)n

µAdSn =

1

2⇡i

Idt0

(t0)nM(0, t0)

[Caron-Huot '17][Alday, Bissi ’16]

Bootstrap in Mellin SpaceWe can also do bootstrap in Mellin space. Consider dispersion relations

M3d(�12, �13, �14)

(�12 � ��3 )(�13 � ��

3 )(�14 � ��3 )

= �M3d(��

3 , �13,2��3 � �13)

(�13 � ��3 )2

1

�12 � ��3

+1

�14 � ��3

!

+1

2

X

⌧,J,m

C2⌧,JQ

⌧,dJ,m(�13)

(�� ⌧2 �m� ��

3 )(�13 � ��3 )(�� 13 + (m+ ⌧

2 ��)� ��3 )

⇥✓

1

�12 �� + ⌧2 +m

+1

�14 �� + ⌧2 +m

◆

Bootstrap in Mellin Space

We now write Polyakov conditions

M(0, �13,�� 13) = 0

The result takes the following formX

⌧,J,m

C2⌧,J↵⌧,J,m = 0

↵⌧,J,m =1

(⌧ � 2��3 + 2m)(⌧ � 4��

3 + 2m)

(⌧ + 2m��)Q⌧,d

J,m(��3 )

(⌧ � 2��3 + 2m)(⌧ � 4��

3 + 2m)� ��

3

Q⌧,dJ,m(��

3 )0

⌧ + 2m� 2��

!.

positive by unitarity

Bootstrap in Mellin Space

The alpha functionals have remarkable properties

1 2 3 4 5 6 7

0.02

0.04

0.06

0.08

0.10

twist

X

⌧,J,m

C2⌧,J↵⌧,J,m = 0scalar operators

Bootstrap in Mellin SpaceThe alpha functionals have remarkable properties

twist

X

⌧,J,m

C2⌧,J↵⌧,J,m = 0spinning operators

2 4 6 8 10 12 140.0

0.5

1.0

1.5

2�� 1.02 1.04 1.06 1.08 1.10

-0.1

0.1

0.2

[Carmi, Caron-Huot ’19][Mazac, Rastelli, Zhou ’19]

They are extremal functionals!

3d Ising

The sum rule takes the form

�X

⌧<2��,J>0,m

C2⌧,J↵⌧,J =

X

⌧>2��,J>0,m

C2⌧,J↵⌧,J +

X

⌧,J=0

C2⌧,J↵⌧,J

Plugging the available numbers in the 3d Ising we get

Holographic Functionals

The whole formalism is particularly suited for holographic theories due to extra suppression of double trace operators.

Let us consider a free scalar in AdS with

which we can try to weakly couple to gravity.

Δ <34

(d − 2)





Δ <34

(d − 2)

Is there a theory of QG that looks like GR+minimal scalar below the Planck scale?





Δ <34

(d − 2)

Is there a theory of QG that looks like GR+minimal scalar below the Planck scale?

Surprisingly, not always!


1.05 1.10 1.15 1.20 1.25�

-5

5

beyond HPPS!

Further comments

Mellin amplitudes are well-defined nonperturbatively

Subtractions (non-Mellin pieces)

“Meromorphic” and polynomially bounded

Positivity of Mack polynomialsDispersion relations in Mellin space (AdS EFT)Matrix elements of light-ray operators

Twist accumulation points

New bootstrap functionals suited for holography

Polyakov conditions

Further comments

Mellin amplitudes are well-defined nonperturbatively

Subtractions (non-Mellin pieces)

“Meromorphic” and polynomially bounded

Positivity of Mack polynomialsDispersion relations in Mellin space (AdS EFT)Matrix elements of light-ray operators

Twist accumulation points

New bootstrap functionals suited for holography

Thank you!

Polyakov conditions

Flat Space Limit[Penedones ’10]

Flat space limit naturally emerges in the fixed angle regime

lims,t!1; st�fixed

M(s, t) ! A(s, t)

[Paulos, Penedones, Toledo, van Rees, Vieira ’16]

Cuts and poles emerge from the condensation of the twist accumulation points.

Why the S-matrix bootstrap is hard?

[from Y.Dokshitzer][Maldacena, Simmons-Duffin, AZ][Gary, Giddings, Penedones]

CFT OPE

Operators can be expanded in the basis of primaries

Oi(x)Oj(0) =X

k

�ijk|x|�i+�j��k (Ok(0) + xµ@µOk(0) + ...)

CFT OPE

Operators can be expanded in the basis of primaries

Oi(x)Oj(0) =X

k

�ijk|x|�i+�j��k (Ok(0) + xµ@µOk(0) + ...)

Consider now the four-point function of identical operators:

hO(x1)O(x2)O(x3)O(x4)i =G(u, v)

(x212x

234)

�

u =x212x

234

x213x

224

, v =x214x

223

x213x

224

CFT Bootstrap2-, 3-point functions are fixed in terms of the OPE data

=

O(x1)

O(x2) O(x3)

O(x4)

O(x1)

O(x2) O(x3)

O(x4)

OiOi

X

i

X

i

CFT data: (�,J) �ijk

Documents

Nonperturbative Mellin Amplitudes: Existence, Properties, … · 2020. 1. 31. · Nonperturbative Mellin Amplitudes: Existence, Properties, Applications A. Zhiboedov (CERN) SISSA,