large spin analytic bootstrap via mellin...large spin analytic bootstrap via mellin parijat dey...

Large spin analytic bootstrap via Mellin

Parijat Dey

CHEP, Indian Institute of Science,

Based on

JHEP 1801(2018) 152 with Kausik Ghosh and Aninda Sinha

Bangalore, India

JHEP1802(2018)153 with Apratim Kaviraj

OIST, OkinawaMarch 20, 2018

Sampling of the results

Anomalous dimension of large spin double trace operators:All orders in inverse spin J

h=d/2, d= dimension

Flashing the results for epsilon expansion

Plan of talk

Plan of talk

● Review of large spin bootstrap

Plan of talk


● Bootstrap in Mellin space

Plan of talk



● Epsilon expansion at large spin

Plan of talk



● Summary

● Epsilon expansion at large spin

Quick Review of conformal bootstrap

s-channel


t-channel

s-channel


If OPE is associative then these two expansion must give the same result

t-channel

s-channel



Obtain all You know everything..!!

t-channel

s-channel



This is the conventional bootstrap program

Obtain all You know everything..!!

t-channel

s-channel


Bootstrap equation

Difficult to solve this equation analytically and get information

Bootstrap equation

Difficult to solve this equation analytically and get informationfor general operators

Bootstrap equation

Difficult to solve this equation analytically and get informationfor general operators

Simplifies in a certain limit...large spin limit and for double traceoperators

Bootstrap equation

Fitzpatrick-Kaplan-Poland-Simmons-Duffin, Komargodski-Zhiboedov

Large spin bootstrap

Leading term

The r.h.s. diverges as v ~ 0.


Leading term


Each term on the l.h.s. goes as


Leading term



Need to sum over infinite large spin double trace operators


Leading term



Need to sum over infinite large spin double trace operators on the left to reproduce the divergence on the r.h.s.


Leading term

Double trace operators


The Unknowns

Anomalous dimension

OPE coefficient



The Unknowns

Anomalous dimension

OPE coefficient

for simplicity



Subleading term of the bootstrap equation




Small v behavior on the rhs : power of v is controlled by the twist




We focus on the minimal twist operators on the rhs


twist spin OPE coeff





twist spin OPE coeff




Expanding the lhs in small u we get

Coefficient of Log u term on the lhs involves the anomalous dimension




Assume the following expansion of the anomalous dimension in large spin limit


Known Mean field OPE coefficient




Known Mean field OPE coefficient



Large spin sum on the lhs can be done at the leading order by converting

Summing over


Summing over


We have from the lhs

Summing over

Focus on the logu term from the rhs



Summing over


Compare the log u term to get the anomalous dimension



Summing over


Compare the log u term to get the anomalous dimension



Similar analysis for non log terms gives the OPE coefficients

Alday-Zhiboedov (2015)


Subleading order corrections can be done following


Change of variable




Anomalous dimension: asymptotic expansion in inverse J

Change of variable





Involves two recursion relations that one needs to solve

Change of variable





Involves two recursion relations that one needs to solve

Simplifies in Mellin space...gives an all order expression !

Change of variable



Conformal bootstrap in Mellin space

Mellin transform

Inverse Mellin transform

Mellin transform


Scaling behavior of CFT correlation function

Mellin transform


Scaling behavior of CFT correlation function

Mellin representation captures conformal symmetry automatically

The Mellin amplitude

Mack (2009), Penedones, Costa-Goncalves- Penedones , Fitzpatrick-Kaplan- Penedones-Raju, van Rees ....

Mellin transform of the 4-pt function



Mellin transform of the 4-pt function

Mellin amplitude



Properties of Mellin amplitude


● Mellin amplitude is a meromorphic functin of (s,t)



● Poles correspond to the primary operators exchanged in



● Poles correspond to the primary operators exchanged in the intermediate states




● Residues tell us about the OPE coefficients




● Residues tell us about the OPE coefficients

● Channel dualities are manifest as s, t exchange

Mellin transform of s channel conformal block

Expand in small anm dim



Residue at double pole

Continuous Hahn polynomial




These are orthogonal polynomials




These are orthogonal polynomials


Play a key role in repackaging the equations in Mellin space


Residue at


Residue at

where


Strategy in Mellin space

Mellin transform of the t channel can be obtained from the s channel by replacing

Idea now is to look at the Mellin transform of the t channel



Idea now is to look at the Mellin transform of the t channel conformal block and expand the t-dependence in terms of



Idea now is to look at the Mellin transform of the t channel conformal block and expand the t-dependence in terms of the continuous Hahn polynomials





We will decompose the bootstrap equation into the basis of




We will decompose the bootstrap equation into the basis of continuous Hahn polynomials



Boils down to simple equations in Mellin space


We will decompose the bootstrap equation into the basis of continuous Hahn polynomials


Mack polynomial, polynomial in s, t

Mellin transform of t channel conformal block

Minimal twist operator

Mack polynomial, polynomial in s, t

Evaluate the residue at


Minimal twist operator

Focus on the log u term


Focus on the log u term

Re express the t dependence in terms of Q(t) polynomials.


t integral has pole at


Bootstrap equation in Mellin space

Algebraic eq !!


Mack polynomial

Algebraic eq !!


Mack polynomial

Need the large spin behavior of continuous Hahn polynoimal.

Algebraic eq !!

Asymptotics of continuous Hahn polynomial

In Mellin space approach the large spin behavior of continuous Hahn


In Mellin space approach the large spin behavior of continuous Hahnpolynomial is the key ingredient



These polynomials can be derived from Wilson polynomials in



These polynomials can be derived from Wilson polynomials in a particular limit




Wilson in 1991 worked out the large argument asymptotics of the Wilson polynomial

James Wilson 1991





Use these results to derive the large spin behavior of continuous

James Wilson 1991





Use these results to derive the large spin behavior of continuous Hahn polynomial

James Wilson 1991



J. L.Fields


PD, K. Ghosh, A. Sinha


Given in terms of generalised Bernoulli polynomial



Given in terms of generalised Bernoulli polynomial

Only even powers of J will appear


In the large spin limit in s channel


From the t channel we have the large spin behavior of the


From the t channel we have the large spin behavior of the continuous Hahn polynomial

In the large spin limit

Comparing powers of J from both sides of the bootstrap equation

in s channel


In the large spin limit

Comparing powers of J from both sides of the bootstrap equation will fix the anomalous dimension at all orders in inverse J

in s channel


Result: Anomalous dimension

Expression for anomalous dimension to all orders in 1/J

Result: Anomalous dimension

Expression for anomalous dimension to all orders in 1/J

e.g. i = 0

Similar expression for OPE coefficient comes from the non log term

Result: OPE coefficient

Given in terms of generalised Bernoulli polynomial and Mack polynomial

Epsilon expansion in the large spin limit

Critical exponents via Epsilon expansion...

● based on Feynman diagrams



● Locate fixed point for which the beta function vanishes



● Locate fixed point for which the beta function vanishes

● Use Callan-Symanzik equation to determine the anomalous dimension



Known up to epsilon^5


Known up to epsilon^5 Kleinert et al


Known up to epsilon^5 Kleinert et al

d=3, N=1

Computing Δ for ϕ2 till 2 loops, in Wilson Fisher fixed point CFT


Computing till 3 loops involves many more diagrams



● OPE coefficients are even more difficult to compute:



● OPE coefficients are even more difficult to compute: involves three point functions

Can bootstrap help?

Results from large spin bootstrap can be used to determine the operator dimension of in the large spin limit

Can bootstrap help?


Remember

Can bootstrap help?


Remember

There are multiple operators with the same minimal twist

Can bootstrap help?


Remember

There are multiple operators with the same minimal twist

Need to sum over

Strategy for epsilon expansion

Step 1. Expand the dimension and OPE in epsilon, OPE coeff can be obtained from Mellin Bootstrap Gopakumar-Kaviraj-Sen- Sinha

Step 2: Use the known results as input

Kleinert et al.Gracey,Manashov et al.

Gopakumar-Kaviraj-Sen- Sinha


Step 2: Use the known results as input

Step 3: Do the sum over

Compute the anomalous dimension of higher spin operators at e^5 in the large spin limit

Kleinert et al.Gracey,Manashov et al.

Gopakumar-Kaviraj-Sen- Sinha


Sampling of new result: Large spin anm. dimensionPD, A. KavirajJHEP1802(2018)153

Subsequent order of anomalous dimension can also be computed using


This gives the anomalous dimension at the order



Similarly, the OPE coefficients can also be computed using



Similarly, the OPE coefficients can also be computed using the large spin OPE results

Summary

Summary

Mellin space techniques can be used to obtain the

Summary

Mellin space techniques can be used to obtain the CFT data for large spin double trace operators to all orders in

Summary

Mellin space techniques can be used to obtain the CFT data for large spin double trace operators to all orders in inverse conformal spin

Summary



Summary


Can be extended for higher twist operators


Summary



Mixed correlator, external operators with spin, supersymmetry


Summary


Relation between the usual bootstrap and Mellin bootstrap


Mixed correlator, external operators with spin, supersymmetry


Thank you

large spin analytic bootstrap via mellin...large spin analytic bootstrap via mellin parijat dey...

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