Conformal Field Theory and the Holographic S-Matrix

Conformal Field Theory
and
The Holographic S-Matrix
A. Liam Fitzpatrick
Stanford University
1007.2412, 1107.1499, 1112.4845, 1208.0337, ....
in collaboration with
Kaplan, Katz, Penedones, Poland, Raju, Simmons-Dufﬁn,
and van Rees
Friday, February 22, 13

Conformal Field Theory
and 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- Lorentz-
invariance: invariance:

n”
tio
ila
“D


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 points
ferromagnets
etc.

Conformal Field
Theories
Classical Gauge Theories
r·E=⇢ r·B=0
@B @E
r⇥E= r⇥B=J+
@t @t

Scale-
invariance

Conformal Field
Particle physics
Theories
Quantum ﬁeld theories are approximately scale-invariant in
between scale boundaries

E.g. The Standard Model
QCD
QED
?
1 18 1 15
mZ ⇠ 10 m ⇤QCD ⇠ 10 m 1
me ⇠ 10 12
m

Conformal Field
Theories
Strongly coupled ﬁxed points
Strongly coupled theories are difﬁcult to study.
Conformal invariance can give us a powerful tool to study
their behavior.

?

Gravity - the Last Force
Gravity at low energies is described by general
relativity.

Gµ⌫ = 8⇡GN Tµ⌫
But at high energies, this description breaks down.

1/2
GN ⇠ Mpl


In 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 Scattering
Our description of high-energy scattering
breaks down

High-energy
collisions make Then they
black holes evaporate through
thermal radiation


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

The Scattering matrix
describes transitions
S
between incoming and
outgoing states.
It is a sharp observable

Quantum dynamics of black holes is an
unresolved question about one of the
fundamental forces.
We should try to understand it!
It is still not known how Hawking’s semi-classical
derivation of information loss is resolved.
But we do have a complete theory of
gravitational dynamics provided by AdS/CFT!


Gravity in AdS/CFT
Gravity in Anti de Sitter Conformal Field Theory
in d+1 dimensions in d dimensions
equivalent!

Scale-
invariance

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

From CFT to Gravity
We 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 space
We want to study gravity in ﬂat space by
“zooming in” to a small region of AdS
AdS 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 Gravity
But it is difﬁcult to see how to take this “ﬂat-
space” limit using the CFT.

?


From CFT to Gravity

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.


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

1) How does the CFT in d-dimensions
describe an S-matrix in d+1?
2) When and why do CFTs have Effective
Field Theory (EFT) descriptions in AdS?


The
Holographic
S-Matrix


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

2

t AdS


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

2

t


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

2

t


By jiggling the CFT in the right way,
you can shoot things from/to this boundary.

This description of the S-matrix is holographic.

2

t


How do we jiggle the CFT to make AdS collisions?

2

? t


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 initial
| final i

The S-Matrix and Momentum
space
Momentum-space amplitudes are functions of Lorentz-
invariant inner products called Mandelstam invariants.

h (p1 ) (p2 ) (p3 ) (p4 )i = f (s, t)
2
Mandelstam s = (p1 + p2 )
invariants 2
t = (p1 + p3 )

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?


CFT Scaling
AdS Energy = Dimension

HAdS = DCFT
AdS Hamiltonian CFT “Dilatation”
Generates time Generates scaling
evolution
HAdS DCFT


The Holographic S-Matrix
So what is momentum space for CFT?

Mellin space


Mellin Amplitude
Like Fourier space, Mellin space is an integral
transform of position space.
h (x1 ) (x2 ) (x3 ) (x4 )i
CFT
Z CFT CFT CFT

⇠ dsdt M (s, t) (x1 x2 ) s
[. . . ]

Mellin variables control scaling exponents


Mellin and the S-Matrix
Correlators at
Scattering at
high energy $ high
scaling dimension

S(s, t) ⇠ M (s, t)
at s, t large 1
(i.e. compared to AdS curvature scale RAdS)
2

Conjectured by
t
Penedones ’10
Proven by
ALF, Kaplan ’11

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 4 is complicated!

4 Z
d
= d xD(x1 x)D(x2 x)D(x3 x)D(x4 x)

4
Fourier Transform
=
But it’s trivial in momentum space!

Comparison to
Momentum Space
Compare to the same example in AdS/CFT
4
Contact interaction LAdS = in AdS
dual CFTx
1 x3 CFT Witten, ’98

AdS $ CFT
4
CFT “lives” on
boundary
x2 x4
CFT CFT
4-point function:
h (x1 ) (x2 ) (x3 ) (x4 )i
CFT CFT CFT CFT

Comparison to
Momentum Space
Compare to the same example in AdS/CFT
4
Contact interaction LAdS = in AdS
CFTx x3 CFT
1
4

x2 x4
CFT CFT

4-point function:
Complicated in Position space
But
M (s, t) = !

Contact Interactions
In standard QFT, local interactions just produce
polynomials in momentum space:

4
(@ ) = 2 2
s +t +u 2

The same thing is true in Mellin space for
contact interactions in AdS!
CFTx x3 CFT
1
✓ ◆
4 1
(@ ) = 2 2 2
s + t + u +O 2
RAdS
x2 x4
CFT CFT

Particle Exchange ALF, Kaplan,
Penedones, Raju,
van Rees, ’11

In standard QFT, particle exchange produces poles,
and Factorization on those poles.
ML MR
1
= ML 2
MR
s m
The same thing is true in Mellin space for
particle exchange in AdS!
CFTx
1 x3 CFT X (m) (m)
M L MR ML MR
= m s 2m
x2 x4
CFT CFT

Feynman Rules
This leads to simple Feynman rules that make the
calculation of tree-level diagrams trivial!
s12 1 5 s45
1 Example: 3 ALF, Kaplan,
Penedones, Raju,
=4 6
5 7 van Rees, ’11

d=4
2 4
Paulos, ’11

2 4 3
1 1 1
M/ +
ure 4: Four-point and ﬁve-point Witten diagrams in cubic scalar theory. +
(s12 4)(s45 4) 3(s12 6)(s45 4) 3(s12 4)(s45 6)
r theories in AdS. Another way of saying this is that when we5add derivative

omial coming from the derivatives at vertices.
+
9(s12 6)(s45 6) .
s, the ‘skeleton diagrams’ with only the propagators are basically just ‘dressed’


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

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


AdS Effective Field
Theory from
Conformal Field
Theory

Structure of EFTs
EFTs have a “gap” in mass between the
“light states” in the theory and the
“heavy states” above the cut-off ⇤
heavy Relevant interactions: less
states important at high energies
⇤
3
Example: µ
light
states 2
µ
e.g.
, e , etc. ⇠
p2

Structure of EFTs
EFTs have a “gap” in mass between the
“light states” in the theory and the
“heavy states” above the cut-off ⇤
heavy Irrelevant interactions: more
states important at high energies
⇤ 4
Example: (@ )
⇤4
light
states 4
p
e.g.
, e , etc. ⇠ 4
cut-off
⇤

Structure of EFTs
EFTs have an expansion in inverse powers of
the cut-off times local interactions.
4 4
(@ ) (@µ @⌫ )
heavy 4
+ 8
+ ...
states ⇤ ⇤
⇤ Scattering amplitudes in this expansion are
polynomials with appropriate powers of ⇤
2
s
light ⇠ 4
+ ...
states ⇤
e.g.
, e , etc. EFT becomes strongly coupled at scale ⇤
and requires new states to restore unitarity.

Effective Conformal Theory
Conformal theories exist with a similar “gap” in
the spectrum of scaling dimensions of operators
This gap can be used as a cut-off in
large
scaling dimensions of operators: we
dimension
operators can “integrate out” operators with
Gap very large scaling dimensions

low
dimension
Example: O=F µ⌫
Fµ⌫ ~ 2
= |E| ~ 2
|B|
operators scaling dimension=4
O

Effective Conformal Theory
Simplest example: an effective CFT with just a
single low-dimension scalar operator O
(and its products and derivatives)
This is a very simple theory. It just
large
dimension describes correlators of O .
operators
e.g. M (s, t) = hOOOOi
Gap
(operators above Gap are not part of
the effective CFT.)
low
dimension Perturbative validity of the
operators theory up to the gap requires
O s
M (s, t) ⇠ #
+ ...
Gap

CFT to AdS
Now, let’s derive the effective ﬁeld
theory in AdS:
Prove that if the Mellin amplitudes of
a CFT have an “EFT-type expansion”
s
M (s, t) ⇠ #
+ ...
Gap
then we can construct an effective ﬁeld
theory in AdS


Mellin and Poles
Mellin amplitudes are meromorphic functions.
(analytic + poles)
Their poles are completely determined by the sum
over states, and vice versa.

X
= ↵
|↵ih↵|

iff 2
|h |↵i|
|↵ih↵|
s ↵

Mellin and Poles
The poles match AdS exchange diagrams!

+ ...
2
|h |↵i|
|↵ih↵| = s ↵
= Poles in s, t + Non-Poles
+ ...
The sum over states also has non-pole contributions

Mellin Amplitude Non-poles
If the non-pole piece in the full Mellin amplitude
has an EFT-type expansion, then we can construct
an AdS effective Lagrangian. ALF, Kaplan ’12
s t
Non-Poles ⇠ ⇤2 + ⇤2 + . . .
M (s, t) = Poles +Polynomial(s, t)

= +
Local EFT
AdS particle exchange interaction

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

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Conformal Field Theory and the Holographic S-Matrix

Similar to Conformal Field Theory and the Holographic S-Matrix (20)

Conformal Field Theory and the Holographic S-Matrix