new approaches to ads amplitudesneo.phys.wits.ac.za/workshop_7/perlmutter.pdf · 2017. 1. 19. ·...

New Approaches to AdS Amplitudes

Eric Perlmutter, Princeton University

The First Mandelstam Theoretical Physics School and Workshop

1/16/17

• AdS perturbation theory = 1/N expansion of CFT correlators

• A lot of interesting gravitational physics is non-perturbative• Black hole processes, tunneling instabilities, topology transitions, …

• But even perturbative physics in AdS is poorly understood!

AdS/CFT: Back to basics

Exhibit A: trees

Even at tree-level – that is, for planar CFT correlators – computations are complicated and somewhat unilluminating.

In computing CFT correlation functions, we prefer to use symmetry as much as possible.

Conformal block: the contribution of an irreducible representation of the (Euclidean) conformal group, SO(d+1,1).

How do we decompose tree-level Witten diagrams into conformal blocks?

How do we decompose tree-level Witten diagrams into conformal blocks?

What is the AdS “dual” of a single conformal block?

Exhibit B: loops

• For technical reasons, almost nothing has been computed!

• A gaping hole in our understanding of AdS amplitudes!

Yes: No:

Why study AdS loops?

1. Amplitudes in curved space

Flat space loop amplitudes contain extremely rich physical and mathematical structures. Given the existence of a flat space limit of AdS (Mellin) amplitudes, these structures should be encoded in, or extended to, the AdS amplitudes.

What is the organizing principle underlying the structure of AdS scattering amplitudes?

Why study AdS loops?

1. Amplitudes in curved space

Flat space loop amplitudes contain extremely rich physical and mathematical structures. Given the existence of a flat space limit of AdS (Mellin) amplitudes, these structures should be encoded in, or extended to, the AdS amplitudes.

What is the organizing principle underlying the structure of AdS scattering amplitudes?

2. The 1/N expansion of CFTs

The definition of a holographic CFT must extend to every order in the 1/N expansion. For instance, if a CFT obeys the RT formula but not the FLM formula, it’s not holographic.

A lot of interesting CFT physics – a-c, double-trace OPE data – hides inside AdS loops.

A way forward

This talk will be split roughly into two parts

1. Trees in AdS• The holographic dual of a conformal block:

• Use this to give a transparent computation of trees in AdS

A way forward

This talk will be split roughly into two parts

1. Trees in AdS• The holographic dual of a conformal block:

• Use this to give a transparent computation of trees in AdS

2. Loops in AdS• A new use for the conformal bootstrap: solve crossing equations in the 1/N

expansion to compute loop amplitudes from the boundary

• Possible in principle, and in practice.

Outline

1. Intro to holographic CFTs and AdS amplitudes

2. Tree-level • Conformal blocks as “Geodesic Witten Diagrams”• Decomposing Witten diagrams

3. One-loop• General form of one-loop Mellin amplitudes• The large c bootstrap• New explicit results for scalar amplitudes

4. Future

hep-th/1508.00501 (w/ E. Hijano, P. Kraus, R. Snively)hep-th/1612.03891 (w/ O. Aharony, F. Alday, A. Bissi)

Based on:

CFT correlators

• We consider a four-point function of a local operator, O(x):

• The “amplitude” has a decomposition into conformal blocks:

• Crossing symmetry imposes the associativity of the OPE:

Cross ratios:

where

Conformal blocks

• Conformal partial waves and conformal blocks are proportional:

Conformal partial wave

Conformal block

• Partial waves with external scalars:• Have series and integral representations

• Are hypergeometric in even d• Are eigenfunctions of an SO(d+1,1) Casimir, [Ferrara, Gatto, Grillo; BPZ;

Zamolodchikov; Dolan, Osborn; Kos,

Poland, Simmons-Duffin; Costa,

Penedones, Poland, Rychkov; …]

• In a large c CFT, the correlator admits a 1/c expansion:

• This is induced by the 1/c expansion of the OPE data. e.g. the double-trace data obey

• At every order in this expansion, the correlator must be crossing-symmetric:

• This is automatically (and magically) implemented in AdS.

Binding energies for AdS two-particle states

Amplitude for bound state formation

Holographic CFT correlators

Holographic CFT correlators

• Loop expansion of AdS Witten diagrams CFT 1/c expansion

How to decompose tree-level Witten diagrams into conformal blocks?

What is the AdS “dual” of a single conformal block?

First, let’s recall the definition of a Witten diagram in pure AdSd+1 for tree-level exchange of a symmetric, traceless spin-𝑙 field:

Propagators

Vertices integrated over AdS

Witten diagram (Indices suppressed)

Ordinary Witten diagrams

Geodesic Witten diagrams• Definition: a geodesic Witten diagram is like an ordinary exchange Witten diagram, but

with vertices integrated over geodesics connecting each pair of boundary points, rather than over all of AdS:

Geodesic Witten diagramWitten diagram

Geodesic Witten diagrams


Pullback of spin-𝑙 bulk-to-bulk propagator to

geodesics


• Holds in any dimension d, for external scalars, with arbitrary masses for all fields.

• Not a conjecture – we proved it!


In the simple case ∆𝟏𝟐= ∆𝟑𝟒= 𝟎, the diagram without external propagators is equal to the conformal block itself.

[Czech, Lamprou, McCandlish, Mosk, Sully; Guica]

This has inspired a recent definition of “geodesic operators,” conformal blocks in kinematic space, and improvements on HKLL bulk reconstruction.

Two Proofs

1. Direct computation: e.g. for 𝑙=0 exchange,

[Ferrara, Gatto, Grillo, Parisi ‘72]!

Two Proofs

1. Direct computation: e.g. for 𝑙=0 exchange,

2. Satisfies the conformal Casimir equation (with correct boundary condition):

• This (partially) explains “why” this works: e.g. for 𝑙=0 exchange,

No support on geodesic diagram!

[Ferrara, Gatto, Grillo, Parisi ‘72]!

Geodesic Witten diagrams rely on symmetries of AdS, not dynamics of AdS/CFT.

However, they become especially useful when computing CFT correlators holographically, via

AdS Witten diagrams.

Outline




4. Future

Goal:

Decompose ordinary Witten diagrams into geodesic Witten diagrams, i.e. conformal partial waves.

0. A geodesic identity in AdSI. Scalar contactII. Scalar exchange

A geodesic identity in AdS• To proceed, we introduce one more identity:

where Known coefficients

Scalar contact diagram

I. Scalar contact

(Apply geodesic identity twice)

(Apply geodesic identity twice)

(Use algebra)

I. Scalar contact

This is the final result:

I. Scalar contact

with the following contributions to squared OPE coefficients

II. Scalar exchange

II. Scalar exchange

This time, there’s also one single-trace

CPW, as expected.

• No integration needed!• Logarithmic singularities (anomalous dimensions) appear trivially.

• Spinning exchange Witten diagrams can be similarly decomposed:

Comments

Outline




4. Future

Loops: prelude

• On CFT side, this is constrained to obey crossing symmetry:

• The plan:1. Show how these amplitudes are constrained by 1/c expansion alone.2. Use crossing symmetry to do better.

• To do so, we’re going to take advantage of the Mellin representation of amplitudes.

Loops: prelude

• A predecessor of our work is that of [Heemskerk, Penedones, Polchinski & Sully].• They consider a tree-level OPE ansatz without cubic couplings:

• This yields (𝐿+2)(𝐿+4)8

solutions of the crossing equations: one for every independent, local quartic bulk vertex with ≤ 2L+2 derivatives.

i.e.

Loops: prelude

• With this ansatz, is simply

expanded in 1/c:

• In the crossing equations, lower-order data acts as a source for higher-order data.

Mellin amplitudes

• Crossing symmetry:

“Mellin amplitude”

( )

[Mack; Penedones;

Fitzpatrick, Kaplan,

Penedones, Raju, van Rees;

Paulos]

Mellin amplitudes @ tree-level


• M(s,t) for exchange diagrams: poles at single-trace twists in exchange diagrams,

• M(s,t) for contact diagrams: polynomials

( )

Twist 𝜏𝑝 + 2n descendants:

[Mack; Penedones;



Paulos]

Mellin amplitudes @ tree-level


• M(s,t) for exchange diagrams: poles at single-trace twists in exchange diagrams,

• M(s,t) for contact diagrams: polynomials

( )

Residues = Mack polynomials (degree l, “level” n) Twist 𝜏𝑝 + 2n

descendants:

[Mack; Penedones;



Paulos]

• The double-trace data is accounted for by the gamma functions. • Double poles are required by 1/c expansion:

• log u anomalous dimensions

• Deriving can be done explicitly using the (known) Mellin representation of .

( )

Mellin amplitudes @ tree-level[Mack; Penedones;



Paulos]

Mellin amplitudes @ one-loop

• In general, M1-loop receives two types of contributions:1. Loop corrections to tree-level data2. New exchanges

• A universal contribution is the 1-loop correction to the [OO] double-trace data

(In the truncated theories like 𝜑4, there are no other terms.)


• In general, M1-loop receives two types of contributions:1. Loop corrections to tree-level data2. New exchanges

• A universal contribution is the 1-loop correction to the [OO] double-trace data

(In the truncated theories like 𝜑4, there are no other terms.)

• We now establish the following simple but powerful claim:

All poles and residues of are fixed by tree-level data.

• Proof:

has simple poles at t=2∆+2n, with residues fixed by .

To fix residues, simply equate these two expressions. e.g. the leading residue is


• This passes various checks (large spin, 𝜑4)

• freg is ambiguous, and sometimes infinite: • Freedom to add a homogeneous solution to the crossing equations• In the bulk, this is the choice of 1-loop renormalization conditions for the quartic part of

the effective action for the light fields.

• We can now derive one-loop data, namely


Enter crossing

Suppose you knew the OPE data by some other means. This is tantamount to knowing the AdS amplitude itself.

Enter crossing


Moreover, using large spin expansion, we can reconstruct the amplitude:

Enter crossing


Moreover, using large spin expansion, we can reconstruct the amplitude:

Given an OPE ansatz, we solve for the anomalous dimensions using the crossing equations at O(1/c2) – at both large and finite spin – thus effectively computing for the dual AdS theory.

Conformal block prerequisite

Before proceeding, let’s review a few more basic properties of conformal blocks.

1. Small u, fixed v = lightcone limit:

2. Small v, fixed u:

Regular near u,v = 0

1-loop crossing

1-loop crossing

• Basic idea:

1. contains a log2(u)log(v) term, that is fixed by tree-level data.

2. By crossing, this implies the existence of a specific log2(v)log(u) term.

1-loop crossing

• Basic idea:


2. By crossing, this implies the existence of a specific log2(v)log(u) term. The relevant log(u) term involves ; matching the two fixes !

1-loop crossing

• Basic idea:


2. By crossing, this implies the existence of a specific log2(v)log(u) term. The relevant log(u) term involves ; matching the two fixes !

• Note: to get the enhanced divergence near v=0, for all spins.

1-loop crossing @ large spin

• Our goal is to solve this equation:

• The enhanced divergence comes from the large spin tail. Let us develop the large spin expansion

• Then if we construct the basis of functions

the solution is

where


To summarize:

1. Compute contribution to from individual [OO]n,s operators for each (n,s) using the above method.

2. Add them up.

We can then extrapolate down to finite 𝑙.


To summarize:

1. Compute contribution to from individual [OO]n,s operators for each (n,s) using the above method.

2. Add them up.

We can then extrapolate down to finite 𝑙.

UV divergences can be read off from the n-scaling of the summand. This depends on the theory through .

Harmonic polylogarithms

• A nice surprise: a certain class of harmonic polylogarithms forms a basis of solutions.

• Definition: for some n-vector , with ai = 0,±1, define the iterated integral

• In our problem, this specific class of HPLs appears: • For integer ∆ ,

• Non-integer ∆ = analytic continuation to non-integer weight HPLs.

• HPLs are ubiquitous in flat space one-loop amplitudes!( )

Outline




4. Future

Example 1: 𝜑4 in AdS

• This is the simplest case: @ tree-level,• What about UV divergences?

• The theory is UV divergent in AdSd+1>3

• In d<7, this is cured by a local 𝜑4 counterterm

• Absent any regularization, we should see divergent spin-0 anomalous dimensions in d>2, but finiteanomalous dimensions for all higher spins (when d<7).

• 1/J expansion: e.g. for d=4, ∆=2,

• Resum down to finite spin:

• Note: these obey , consistent with the lightcone bootstrap.

d = 2: d = 4:



• Now let’s reconstruct the full amplitude:

This matches a previous bulk calculation![Penedones;

Fitzpatrick,

Kaplan]



• Now let’s reconstruct the full amplitude:

This matches a previous bulk calculation![Penedones;

Fitzpatrick,

Kaplan]

(One can also analytically compute finite-spin anomalous dimensions directly from M1-loop; they match the results from crossing.)


• The one-loop diagrams include the triangle, which has never been computed:

Example 2: 𝜑3 − 𝜑4 in AdS

• The one-loop diagrams include the triangle, which has never been computed.

• For ∆ = 2, we find:


d = 2: d = 4:


• This yields the four-point triangle Witten diagram: in the t-channel, for instance,

• Would be exciting to reproduce by direct bulk calculation!

d = 2:

d = 4:

Summary of results

• Geodesic Witten diagrams geometrize CFT conformal blocks, and provide a simple way to decompose tree-level AdS Witten diagrams.

• Loop-level AdS amplitudes may be computed via holography from the large c conformal bootstrap.

• We performed explicit one-loop calculations, including a new diagram that has never been computed.

Future directions

• Immediate goals:

o Add gravity = Add to the crossing equations

o Non-planar correlators in full-fledged CFTs: N=4 SYM, (2,0) CFT

• Lofty goals:

o What is the AdS analog of flat space amplitude technology?

o Use the conformal bootstrap to address higher-loop questions in flat space:i. Above what dimension does maximal SUGRA diverge at five loops?ii. What is the string moduli dependence of the D8R4 counterterm in type IIB string theory?

new approaches to ads amplitudesneo.phys.wits.ac.za/workshop_7/perlmutter.pdf · 2017. 1. 19. ·...

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