Conformal Field Theory and the Holographic S-Matrix

Conformal Field Theory and

The Holographic S-MatrixA. Liam FitzpatrickStanford University

in collaboration withKaplan, Katz, Penedones, Poland, Raju, Simmons-Duffin,

and van Rees

1007.2412, 1107.1499, 1112.4845, 1208.0337, ....

Friday, February 22, 13

Conformal Field Theoryand Gravity


Outline

• Conformal Field Theories (CFTs)

• Incompleteness of Gravity at High Energies

• How do CFTs describe gravitational scattering?

• When are CFTs described by Effective Field Theories of gravity?


Conformal Invariance

Conformal = Scale-invariant + Lorentz-invariant

Scale-invariance:

Lorentz-invariance:

“Dilation”


Conformal Field Theories

Conformal Field Theories are relevant for describing a wide range of phenomena.

phase transitions and critical exponents

E.g. Ising model

also: liquid-gas critical pointsferromagnets

etc.



Classical Gauge Theories

Scale-invariance

r ·E = ⇢ r ·B = 0

r⇥E = �@B

@tr⇥B = J+

@E

@t



Quantum field theories are approximately scale-invariant in between scale boundaries

Particle physics

E.g. The Standard Model

QEDQCD

?m�1

Z ⇠ 10�18m ⇤�1QCD ⇠ 10�15m m�1

e ⇠ 10�12mFriday, February 22, 13


Conformal invariance can give us a powerful tool to study their behavior.

Strongly coupled fixed points

Strongly coupled theories are difficult to study.

?Friday, February 22, 13

Gravity - the Last ForceGravity at low energies is described by general

relativity.

Gµ⌫ = 8⇡GNTµ⌫

G�1/2N ⇠ Mpl

But at high energies, this description breaks down.


Gravity - the Last ForceIn contrast to gauge theories, quantizing gravity

at high energies is notoriously hard.Quantum behavior of black holes is still not understood.Hawking evaporation is not unitary: information is lost!


Gravitational ScatteringOur description of high-energy scattering

breaks down

Then they evaporate through thermal radiation

High-energy collisions make

black holes


The Scattering matrix describes transitions

between incoming and outgoing states.

It is a sharp observable

SThe Scattering matrix describes transitions

between incoming and outgoing states.

It is a sharp observable

S

S-Matrix and GravityWe want a theory that describes scattering at any energy.


Gravity - the Last Force

It is still not known how Hawking’s semi-classical derivation of information loss is resolved.

Quantum dynamics of black holes is an unresolved question about one of the

fundamental forces.We should try to understand it!

But we do have a complete theory of gravitational dynamics provided by AdS/CFT!


Gravity in AdS/CFTGravity in Anti de Sitter

in d+1 dimensionsConformal Field Theory

in d dimensions

Scale-invariance

So studying CFTs teaches us about gravity, and vice versa!

equivalent!


From CFT to GravityWe can take known CFTs and answer any

question about quantum gravity, including at high energies.

This description of gravitational scattering is calculated in the CFT, and is “holographic”.


AdS vs. flat spaceWe want to study gravity in flat space by

“zooming in” to a small region of AdSAdS is hyperbolic:

“Flat-space limit of AdS” is the limit of physics on scales much smaller than the AdS radius of curvature.


From CFT to GravityBut it is difficult to see how to take this “flat-

space” limit using the CFT.

?


Before our work, it was not sharply understood how a CFT describes a flat-space gravitational S-matrix.

Despite its importance, the “holographic” equivalence between d-dimensional CFTs and

(d+1)-dimensional gravity theories has many open questions.

From CFT to Gravity


AdS/CFT Questions

2) When and why do CFTs have Effective Field Theory (EFT) descriptions in AdS?

1) How does the CFT in d-dimensions describe an S-matrix in d+1?

In the rest of this talk, I will show you how we have answered the following concrete questions:


The Holographic

S-Matrix


The S-Matrix and Anti-de SitterAdS is a very special box

t

� �

2

AdS


t

� �

2

The S-Matrix and Anti-de Sitter

Infinite in size, but curved geometry lets light travel to infinity and back in finite time


t

� �

2


So it has a boundary.This is where the dual CFT lives.


t

� �

2

The S-Matrix and Anti-de SitterBy jiggling the CFT in the right way,

you can shoot things from/to this boundary.

This description of the S-matrix is holographic.


t

� �

2


How do we jiggle the CFT to make AdS collisions?

?Friday, February 22, 13

The S-Matrix and Momentum space

How do we do this in quantum field theory in flat space?Calculate scattering amplitudes using correlation

functions in momentum space.

h�(p1)�(p2)�(p3)�(p4)i

h i|initial final


The S-Matrix and Momentum space

h�(p1)�(p2)�(p3)�(p4)i =s = (p1 + p2)

2

t = (p1 + p3)2

Mandelstam invariants

f(s, t)

Momentum-space amplitudes are functions of Lorentz-invariant inner products called Mandelstam invariants.


Momentum space for CFTs?

We want a set of coordinates like momentum space that makes it easy to

obtain the holographic S-matrix.

We already have some guidance from AdS/CFT. What is the CFT dual of AdS

frequencies?


AdS Energy

CFT “Dilatation”AdS HamiltonianGenerates scaling

DCFT

Generates scaling

DCFT

Generates time evolution

HAdS

Generates time evolution

HAdS

CFT Scaling Dimension=

HAdS = DCFT


The Holographic S-Matrix

So what is momentum space for CFT?

Mellin spaceMellin space


Mellin Amplitude

h�(x1)�(x2)�(x3)�(x4)iCFT CFT CFT CFT

M( , t)s

Mellin variables control scaling exponents

(x1 � x2) [. . . ]s

Like Fourier space, Mellin space is an integral transform of position space.

⇠Z

dsdt


at large (i.e. compared to AdS curvature scale )

Mellin and the S-Matrix

$Scattering at high energy

Correlators at high

scaling dimension

Conjectured by Penedones ’10

Proven by ALF, Kaplan ’11

t

� �

2

t

� �

2

S(s, t) ⇠ M(s, t)s, t

R�1AdS


Mellin and Calculations

Just like momentum space, Mellin space is extremely useful for doing calculations

The calculations are easier, and the results are much simpler to understand


Comparison to Momentum Space

Consider standard QFT.In position space, even is complicated!��4

=

Zd

dxD(x1 � x)D(x2 � x)D(x3 � x)D(x4 � x)

��4

But it’s trivial in momentum space!

Fourier Transform

��4

=�But it’s trivial in momentum space!

Fourier Transform

��4

=�Friday, February 22, 13

�CFT

�CFT �CFT

�CFTx1

x2

x3

x4

��4

CFT “lives” on boundary

4-point function:

h�(x1)�(x2)�(x3)�(x4)iCFT CFT CFT CFT

Contact interaction in AdSLAdS = ��4Compare to the same example in AdS/CFT


�AdS $ �CFT

dual Witten, ’98


Complicated in Position space

�CFT

�CFT �CFT

�CFTx1

x2

x3

x4

��4

4-point function:

Contact interaction in AdSLAdS = ��4Compare to the same example in AdS/CFT


But !M(s, t) = �Friday, February 22, 13

Contact InteractionsIn standard QFT, local interactions just produce

polynomials in momentum space:

=(@�)4

The same thing is true in Mellin space for contact interactions in AdS!

�CFT

�CFT �CFT

�CFTx1

x2

x3

x4

= s2 + t2 + u2 +O✓

1

R2AdS

◆(@�)4

The same thing is true in Mellin space for contact interactions in AdS!

�CFT

�CFT �CFT

�CFTx1

x2

x3

x4

= s2 + t2 + u2 +O✓

1

R2AdS

◆(@�)4

s2 + t2 + u2


Particle Exchange

ML MR

In standard QFT, particle exchange produces poles,and Factorization on those poles.

= 1

s�m2ML MR

ALF, Kaplan, Penedones, Raju,

van Rees, ’11

ML MR

The same thing is true in Mellin space for particle exchange in AdS!

�CFT

�CFT

x1

x2 �CFT

�CFTx3

x4

=X

m

M (m)L M (m)

R

s�� 2m

ML MR

The same thing is true in Mellin space for particle exchange in AdS!

�CFT

�CFT

x1

x2 �CFT

�CFTx3

x4

=X

m

M (m)L M (m)

R

s�� 2m


1

2

3

4

5

1

2

3

4

5

6 7

Figure 4: Four-point and five-point Witten diagrams in cubic scalar theory.

to all scalar theories in AdS. Another way of saying this is that when we add derivative

interactions, the ‘skeleton diagrams’ with only the propagators are basically just ‘dressed’

by a polynomial coming from the derivatives at vertices.

4 Sample Computations

In this section we will demonstrate the power of our formalism by computing the 5-pt and

6-pt amplitudes in a scalar field theory with 3-pt interaction vertices. Notice that, as will

become clear below, using the factorization formula it is even easier to compute amplitudes

in theories with general ⌃a⌅b vertices, since the greatest complication arises from having

many bulk to bulk propagators.

Before moving on to a non-trivial computation, let us see how our formalism works in the

simplest case, that of the 4-pt function. Suppose specifically that we have the bulk interaction

vertices ⇥⌅1⌅2⌅5 and ⇥⌅3⌅4⌅5, and we want the Mellin amplitude for ⌅O1O2O3O4⇧ from ⌅5

exchange, as shown in Fig. 4. Applying equation (67), we find the Mellin amplitude is

M4(�ij) =⇧

m

1

�LR �⇥5 � 2m

�4⇤h�(⇥5 � h+ 1)m!

(⇥5 � h+ 1)m

⇤�⇥125

(�12)mm!

⇥�⇥345

(�34)mm!

⇥⌅

�LR=�+2m

=⇧

m

1

�5 �m

�2⇤h�(⇥5 � h+ 1)m!

(⇥5 � h+ 1)m

�⇥125

1

m!(⇥12,5)�m

⇥�⇥345

1

m!(⇥34,5)�m

⇥, (73)

where ⇥ij,k ⇥ �i+�j��k

2 and ⇥ijk is the 3-pt Mellin amplitude for a contact Witten diagram

with external dimensions ⇥i, ⇥j and ⇥k. In the second line, we have used the fact that

2�12 = ��LR + ⇥1 + ⇥2, 2�34 = ��LR + ⇥3 + ⇥4, and the identity (a � m)m = 1(a)�m

. We

24

Feynman RulesThis leads to simple Feynman rules that make the

calculation of tree-level diagrams trivial!s12 s45Example:

d = 4

�� = 4

M / 1

(s12 � 4)(s45 � 4)

ALF, Kaplan, Penedones, Raju,

van Rees, ’11

Paulos, ’11

+1

3(s12 � 6)(s45 � 4)+

1

3(s12 � 4)(s45 � 6)

+5

9(s12 � 6)(s45 � 6).Friday, February 22, 13

AdS/CFT Questions




AdS Effective Field Theory from

Conformal Field Theory


Structure of EFTsEFTs have a “gap” in mass between the

“light states” in the theory and the “heavy states” above the cut-off

⇤

⇤heavy states

light statese.g.� e�, , etc.

Relevant interactions: less important at high energies

Example: µ�3

� �

� �� ⇠ µ2

p2Friday, February 22, 13

Structure of EFTsEFTs have a “gap” in mass between the

“light states” in the theory and the “heavy states” above the cut-off

⇤

⇤heavy states


Irrelevant interactions: more important at high energies

� �

� �⇠

(@�)4

⇤4

p4

⇤4cut-off

Example:


Structure of EFTsEFTs have an expansion in inverse powers of

the cut-off times local interactions.

⇤

heavy states


Scattering amplitudes in this expansion are polynomials with appropriate powers of

⇠� �

� �

(@�)4

⇤4+

(@µ@⌫�)4

⇤8+ . . .

⇤

s2

⇤4+ . . .

EFT becomes strongly coupled at scaleand requires new states to restore unitarity.

⇤


Effective Conformal TheoryConformal theories exist with a similar “gap” in the spectrum of scaling dimensions of operators

large dimension operators

low dimension operatorsO

�Gap

This gap can be used as a cut-off in scaling dimensions of operators: we can “integrate out” operators with

very large scaling dimensions

O= Fµ⌫Fµ⌫ = | ~E|2 � | ~B|2Example:

scaling dimension=4


Effective Conformal TheorySimplest example: an effective CFT with just a

single low-dimension scalar operator (and its products and derivatives)

OThis is a very simple theory. It just

describes correlators of .Olarge dimension operators

low dimension operatorsO

�Gap M(s, t) = hOOOOie.g.

(operators above are not part of the effective CFT.)

�Gap

Perturbative validity of the theory up to the gap requires

M(s, t) ⇠ s

�#Gap

+ . . .


CFT to AdSNow, let’s derive the effective field

theory in AdS:Prove that if the Mellin amplitudes of a CFT have an “EFT-type expansion”

then we can construct an effective field theory in AdS

M(s, t) ⇠ s

�#Gap

+ . . .


Mellin and PolesMellin amplitudes are meromorphic functions.

� �

� �

=X

↵

�

�

�

�h↵||↵i

Their poles are completely determined by the sum over states, and vice versa.

�

�

�

�h↵||↵i |h��|↵i|2

s��↵

iff

(analytic + poles)


= +Poles

in s, t

+

The poles match AdS exchange diagrams!

. . .

�

�

�

�h↵||↵i |h��|↵i|2

s��↵= + . . .

Non-Poles

The sum over states also has non-pole contributions

Mellin and Poles


=

=

Poles

+

M(s, t)

AdS particle exchangeLocal EFT interaction

+Polynomial(s, t)

ALF, Kaplan ’12

Mellin Amplitude Non-polesIf the non-pole piece in the full Mellin amplitude has an EFT-type expansion, then we can construct

an AdS effective Lagrangian.

Non-Poles

s

⇤2+

t

⇤2+ . . .⇠


AdS/CFT Questions




Future Directions• Find CFT description of black hole

formation and evaporation• Feynman Rules for general loop

diagrams and particles with spin• Use Mellin space to describe dS/CFT

• Study CFT interpretation of Modified Theories of Gravity in AdS

• Understand bulk EFT for broken conformal invariance (QCD)


The End


Conformal Field Theory and the Holographic S-Matrix

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