Momentum space approach to crossing symmetric CFT correlators - - PowerPoint PPT Presentation

Momentum space approach to crossing symmetric CFT correlators

Hiroshi Isono (Chulalongkorn University)

in collaboration with

Toshifumi Noumi (Kobe) Gary Shiu (Wisconsin-Madison)

based on 1805.11107, 1903.01110, 1908.04572

Toshiaki Takeuchi (Kobe)

a novel basis of four-point conformal correlators topic

Introduction

CFT in general dimensions

CFT = analyse solutions of the conformal WT identities the solution space of WT identities is a vector space finding a good basis is crucial. this talk focuses on the basis for 4pt functions

translation, rotation, dilatation, special conformal transformation

4pt functions

crossing symmetry consistency with operator product expansion (OPE) invariance under (1234) → (1324), (1423)

O1 × O2 ∼ ∑

φ

cφ(x12)αφφ ⟷ ⟨O1O2O3O4⟩ ∼ ∑

φ

cφ(x12)αφ⟨φO3O4⟩

⟨O1O2O3O4⟩

4pt blocks

1) crossing symmetry 4pt blocks satisfying both 1) and 2) have not been found 2) consistency with operator product expansion (OPE) 0) conformal covariance it is natural to require that 4pt basis should satisfy

4pt blocks

two classes of the 4pt block have been known conformal block consistency with OPE manifest crossing symmetry explicitly broken Polyakov block consistency with OPE obscured crossing symmetry manifest (1974 Polyakov) GO(12; 34)

⟨O1O2O3O4⟩ = ∑

O

1 2 3 4 O 1 2 3 4 O

WO(1234)

⟨O1O2O3O4⟩ = ∑

O

1 2 3 4 O 1 2 3 4 O

( (

+ +

(1973 Ferrara Grillo Gatto) constraint by hand constraint by hand

But he didn’t give an explicit form of the basis in momentum. Instead, he presented its position-space version. We derived explicit forms of Polyakov block in momentum space

Polyakov block

Polyakov gave a characterising definition of the crossing-sym. basis in momentum space. What we did

This talk will focus on the definition and construction of Polyakov block.

[1805.11107, 1908.04572]

Definition and Construction

Polyakov block

The definition of Polyakov block is based on the analytic structure

• f 2 and 3-point functions in momentum space.

Polyakov block: 3pt function

3pt function in momentum space

(Fourier transf. or Solve WT, NO holography used) Δ = d

2+ν

bulk-boundary propagator in AdS

∝ zd/2−ν [−(pz)νIν(pz)+(pz)νI−ν(pz)]

⟨O1(p1)O2(p2)O3(p3)⟩′ = C123∫

∞

dz zd+1 ℬν1(p1, z)ℬν2(p2, z)ℬν3(p3, z)

1974 Ferrara Gatto Grillo Parisi

where

ℬν(p, z) =

1 2ν−1Γ(ν) pνzd/2Kν(pz)

Polyakov block: 3pt function

where

ℬν(p, z) =

1 2ν−1Γ(ν) pνzd/2Kν(pz) ∝ zd/2−ν [−(pz)νIν(pz)+(pz)νI−ν(pz)]

non-analytic in p2

Discp2

3⟨O1(p1)O2(p2)O3(p3)⟩′ = T12;3(p1, p2; p3) Discp2 3⟨O3(−p3)O3(p3)⟩′

Non-analytic part

Δ = d

2+ν

Iν(pz) ∼ (pz)ν[1 + 𝒫((pz)2)]

⟨O(p)O(−p)⟩ ∝ p2ν

(pz)νIν(pz) = z2ν[1 + 𝒫((pz)2)] × p2ν

= [analytic in p2] × ⟨O(p)O(−p)⟩

cubic vertex analytic in p2

3

analytic in p2

3pt function in momentum space

⟨O1(p1)O2(p2)O3(p3)⟩′ = C123∫

∞

dz zd+1 ℬν1(p1, z)ℬν2(p2, z)ℬν3(p3, z)

(Fourier transf. or Solve WT, NO holography used)

1974 Ferrara Gatto Grillo Parisi

factorisation

Polyakov block: analogy

Discp2

3⟨O1(p1)O2(p2)O3(p3)⟩′ = T12;3(p1, p2; p3) Discp2 3⟨O3(−p3)O3(p3)⟩′

# analogous to the case of the flat-space 3pt vertex

p1 p2

×

=

p1 p2

this may invoke the following factorisation for 4pt amplitudes (in s-channel)

p1 p2 p4

p3

×

×

p1 p4

p3 p2

Discp2

12=−m2

Discp2

12=−m2

=

Polyakov adopted a similar factorisation in the context of CFT

p12 p12

Discp2

12=−m2

Discp2

12=−m2

Polyakov block: definition

introduce

W(s)

O , W(t) O , W(u) O

and for a primary field O,

Discp2

12W(s)

O = T12;O(p1, p2; − p12) Discp2

12⟨O(p12)O(−p12)⟩ T34;O(p3, p4; p12)

(1) For a 4pt function ⟨O1O2O3O4⟩

p12 = p1 + p2

(2) no other non-analyticity than the above one for W(s)

O

non-analyticity in the intermediate momentum For the s-channel Polyakov block W(s)

O

⟨O1O2O3O4⟩ = ∑

O

(W(s)

O + W(t) O + W(u) O )

their sum is crossing-sym.

p1 p2 p4

p3

×

×

p1 p4

p3 p2

Discp2

12

Discp2

12

=

Discp2

12W(s)

O = T12;O(p1, p2; − p12) Discp2

12⟨O(p12)O(−p12)⟩′ T34;O(p3, p4; p12)

Polyakov block: construction

Discp2

12W(s)

O = T12;O(p1, p2; − p12) Discp2

12⟨O(p12)O(−p12)⟩′ T34;O(p3, p4; p12)

W(s)

O = T12;O(p1, p2; − p12) ⟨O(p12)O(−p12)⟩′ T34;O(p3, p4; p12)

Polyakov block: construction

1st trial

T12;O ∼ ∫

∞

dz zd+1 K(p1z)K(p2z)I(p12z)

diverges when p1 + p2 = p12

Kν(x) ∼ e−x, Iν(x) ∼ ex as x ∼ ∞

p1 p2 p1 p12

1) satisfying the factorisation condition 2) but it has another non-analyticity

1st trial 1) satisfying the factorisation condition

Discp2

12W(s)

O = T12;O(p1, p2; − p12) Discp2

12⟨O(p12)O(−p12)⟩′ T34;O(p3, p4; p12)

2nd trial

no such divergence!

T12;O ∼ ∫

∞

dz zd+1 K(p1z)K(p2z)I(p12z)

I12;O(p1, p2; − p12) ⟨O(p12)O(−p12)⟩′ I34;O(p3, p4; p12)

Therefore satisfies the 2nd.

However, its discontinuity contains not only

but also its shadow

Disc . ⟨OO⟩ ∼ p2ν

W(s)

O = T12;O(p1, p2; − p12) ⟨O(p12)O(−p12)⟩′ T34;O(p3, p4; p12)

Disc . ⟨ ˇ O ˇ O⟩ ∼ p−2ν

the factorisation NOT satisfied!

diverges when p1 + p2 = p12

Polyakov block: construction

I12;O ∼ ∫

∞

dz zd+1 K(p1z)K(p2z)K(p12z)

Kν(x) ∼ e−x, Iν(x) ∼ ex as x ∼ ∞

p1 p2 p1 p12

2) but it has another non-analyticity

Polyakov block: construction

How to find a way out? — block the divergence of the other

∫

∞

dz zd+1 K(k1z)K(k2z)I(k12z) at z ∼ ∞

by reducing the integration range

[0, z′]

— For factorisation, replace just one of the two by — Lesson from the 2nd trial : replacing both cubic vertices is too much

∫

∞

dz zd+1 K(k1z)K(k2z)K(k12z)

[0, ∞] →

Polyakov block: construction

How to find a way out? — block the divergence of the other

∫

∞

dz zd+1 K(k1z)K(k2z)I(k12z) as z ∼ ∞

by reducing the integration range

[0, z′]

— For factorisation, replace just one of the two by Answer

∫

∞

dz zd+1 dz′ z′d+1 ℬν1(k1, z)ℬν2(k2, z)

× [ θ(z − z′)Kν(k12z)Iν(k12z′) + θ(z′− z)Kν(k12z′)Iν(k12z) ] × ℬν3(k3, z′)ℬν4(k4, z′)

the step function blocks the divergence produce only Disc . ⟨OO⟩ ∼ k2Δ

— Lesson from the 2nd trial : replacing both cubic vertices is too much

as z ∼ ∞

∫

∞

dz zd+1 K(k1z)K(k2z)K(k12z)

(the intermediate operator O is scalar)

[0, ∞] →

W(s)

O = ∫ ∞

dz zd+1 dz′ z′d+1 ℬν1(k1, z)ℬν2(k2, z)

× [ θ(z − z′)Kν(k12z)Iν(k12z′) + θ(z′− z)Kν(k12z′)Iν(k12z) ] × ℬν3(k3, z′)ℬν4(k4, z′)

This is nothing but an exchange Witten diagram. NB: contact diagrams like are analytic Therefore, it satisfies the discontinuity condition trivially, so they can be added freely to .

W(s)

O

• cf. Gopakumar et al (2016) in position space as a purely mathematical coincidence.

Polyakov block: construction

• cf. exchange interaction in dS for cosmological collider (G.L. Pimentel’s talk)

the crucial step is to find an analytic form of the 3pt vertex

discp2

12

p1 p2

=

×

discp2

12 ⟨OΔOΔ⟩

⟨O1O2OΔ⟩

we found analytic forms of ⟨O1O2OΔ⟩ in momentum space by solving conformal WT identities 1805.11107, 1908.04572 : scalar-scalar-spin s 1903.01110, 1909.***** : conserved currents-spin s

Remark 1: 3pt functions

See Toshiaki Takeuchi’s poster for details

three scalars: Ferrara-Gatto-Grillo-Parisi (4d), Bzowski-McFadden-Skenderis (any dim.) scalars+currents (spin 2): Bzowski-McFadden-Skenderis

≤

[HI-Noumi-Shiu] [HI-Noumi-Takeuchi]

Polyakov block can separate the Gaussian part from the connected (interacting) part thanks to the crossing symmetry

Remark 2

Polyakov block for with the intermediate identity operator

⟨O1O2O3O4⟩

Widentity = ⟨O1O2⟩⟨O3O4⟩ + ⟨O1O3⟩⟨O2O4⟩ + ⟨O1O4⟩⟨O2O3⟩ nothing but the disconnected (Gaussian) part of .

⟨O1O2O3O4⟩

⟨O1O2O3O4⟩connected = ∑

X≠identity

WX

• cf. s-channel conformal block

⟨O1O2⟩⟨O3O4⟩ + ⟨O1O3⟩⟨O2O4⟩ + ⟨O1O4⟩⟨O2O3⟩ GX(12; 34) X = identity double trace O∂2nO

• application to inflationary correlators

Summary

(cosmological collider, …)

• combination with development of scattering theory

up to analytic ones such as contact Witten diagrams We derived explicit forms of Polyakov block in momentum space and showed that it is the exchange Witten diagram analyticity, causality; counterpart of Froissart bound? …

Outlook

Thank you very much!