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
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?
Friday, February 22, 13
Conformal Invariance
Conformal = Scale-invariant + Lorentz-invariant
Scale-invariance:
Lorentz-invariance:
“Dilation”
Friday, February 22, 13
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.
Friday, February 22, 13
Conformal Field Theories
Classical Gauge Theories
Scale-invariance
r ·E = ⇢ r ·B = 0
r⇥E = �@B
@tr⇥B = J+
@E
@t
Friday, February 22, 13
Conformal Field Theories
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 Field Theories
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.
Friday, February 22, 13
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!
Friday, February 22, 13
Gravitational ScatteringOur description of high-energy scattering
breaks down
Then they evaporate through thermal radiation
High-energy collisions make
black holes
Friday, February 22, 13
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.
Friday, February 22, 13
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!
Friday, February 22, 13
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!
Friday, February 22, 13
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”.
Friday, February 22, 13
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.
Friday, February 22, 13
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.
Friday, February 22, 13
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.
Friday, February 22, 13
From CFT to GravityBut it is difficult to see how to take this “flat-
space” limit using the CFT.
?
Friday, February 22, 13
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
Friday, February 22, 13
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:
Friday, February 22, 13
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
Friday, February 22, 13
t
� �
2
The S-Matrix and Anti-de Sitter
So it has a boundary.This is where the dual CFT lives.
Friday, February 22, 13
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.
Friday, February 22, 13
t
� �
2
The S-Matrix and Anti-de Sitter
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
Friday, February 22, 13
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.
Friday, February 22, 13
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?
Friday, February 22, 13
AdS Energy
CFT “Dilatation”AdS HamiltonianGenerates scaling
DCFT
Generates scaling
DCFT
Generates time evolution
HAdS
Generates time evolution
HAdS
CFT Scaling Dimension=
HAdS = DCFT
Friday, February 22, 13
The Holographic S-Matrix
So what is momentum space for CFT?
Mellin spaceMellin space
Friday, February 22, 13
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
Friday, February 22, 13
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
Friday, February 22, 13
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
Friday, February 22, 13
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
Comparison to Momentum Space
�AdS $ �CFT
dual Witten, ’98
Friday, February 22, 13
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
Comparison to Momentum Space
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
Friday, February 22, 13
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
Friday, February 22, 13
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
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?
Friday, 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
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
light statese.g.� e�, , etc.
Irrelevant interactions: more important at high energies
� �
� �⇠
(@�)4
⇤4
p4
⇤4cut-off
Example:
Friday, February 22, 13
Structure of EFTsEFTs have an expansion in inverse powers of
the cut-off times local interactions.
⇤
heavy states
light statese.g.� e�, , etc.
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.
⇤
Friday, February 22, 13
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
Friday, February 22, 13
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
+ . . .
Friday, February 22, 13
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
+ . . .
Friday, February 22, 13
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)
Friday, February 22, 13
= +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
Friday, February 22, 13
=
=
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+ . . .⇠
Friday, February 22, 13
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?
Friday, February 22, 13
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)
Friday, February 22, 13