Spinning Witten Diagrams: From Geodesic to Mellin · 2020. 6. 24. · Spinning Witten Diagrams:...

Spinning Witten Diagrams: From Geodesic to Mellinwith En-Rui Kuo and Hideki Kyono, JHEP 1705 (2017) 079 +1712.07991

Heng-Yu ChenNational Taiwan University

OIST Bootstrap Symposium, March 2018

Heng-Yu Chen National Taiwan University Spinning Witten Diagrams: From Geodesic to Mellin

In a d-dim. CFT, 1, 2 3 point correlations for the primary operators{O∆i ,li (xi )} are fixed by conformal symmetries, 4 point is the firstnon-trivial one:

where Oi = O∆i ,0, xij = xi − xj , ∆ij = ∆i −∆j , F (u, v) is a function of:

Decomposing F (u, v) into different OPE channels (say s-channel):

where λ12O and λ34O are OPE coefficients, G∆,J(u, v) and W∆,J(xi ) arethe “conformal block” and “conformal partial wave” for O∆,J family.


I The scalar conformal block G∆,J(u, v) needs to satisfy the quadraticCasimir equation” [Dolan-Osborn 2003, 2011]

for even d , G∆,J(u, v) is given by hypergeometric functions.

I Generally, the conformal partial wave W∆,J(xi ) has the followingintegral representation:

where Od−∆,J(x0) is the “shadow” operator and integration kernal:

This is sometimes called “spectral/split” representation of CPW.


For CFTd with weakly coupled AdSd+1 dual, e.g. N = 4 SYM, or moregeneral large N CFTs, we usually divide {O∆,J} into: [D‘Hoker-Freedman 2002]

where ∆ can receive quantum corrections. In 1/N counting, they giveleading the contributions to correlation functions.In addition at some sub-leading 1/N, we have contributions from:

where ∆(ij)(n, J) = ∆i + ∆j + 2n + J + γij(n, J).

where P(12)n,J and P

(34)m,J are products of OPE coefficients for (12) and (34)

channels.


The scalar four point function is also known to enjoy following Mellinrepresentation (Mack, Penedones, Fitzpatrick, Kaplan, Raju, Van Rees, Paulos +....)

Here the integrand M(s, t) is called “Mellin Amplitude” (or reduced?):

with kinematical Mack polynomial Pν,J(s, t) and {δij} are:

We have clean separation among the single trace and double traceoperators ∼ O∆1∂

J(∂2)mO∆2 in Mellin representation, in particularM(s, t) is only fixed by the spectrum of single trace operators.


Mellin amplitude M(s, t) shares many similarities with flat space QFTscattering amplitudes, in particular, introducing fictitious momenta {pi}:

We have four constraints, hence two independent Mandelstam variables(s, t) (Can also have u =

∑4i=1 ∆i − (s + t)):

We can now consider expand in either of (s, t, u) variable, say:

where the infinite poles are precisely located at O∆,J with twistτ = ∆− J and its descendants (∂2)mO∆,J .


We also consider spinning primary O∆i ,li , transforming in symmetrictraceless rep., their 3 point function is again fixed by conformalsymmetries: [Costa et al 2011]

Here (Pi ,Zi ) are the embedding coordinates and polarization vectors:

where


The primary operator O∆i ,li (Pi ,Zi ) is a degree li polynomial in Zi

where O∆i ,A1...Ali(Pi ) is a symmetric traceless transverse (STT) tensor.

Here we have also defined the “box basis” for spinning 3 point functions:

where Pij = −2Pi · Pj , τi = ∆i + li and tensor structures are encoded in:

Recovering the tensor indices using differential operator DZi .


I The triplet of non-negative integers {n12, n13, n23} need to satisfy:

and these partitions label the possible resultant tensor structures.

I Another useful tensor basis is the “differential basis”:

where the shift operator Σa,b : (∆1,∆2)→ (∆1 + a,∆2 + b) andD11,D12,D22 and D21 only involve P1,2 and Z1,2:


I In differential basis, we explicitly break the permutation symmetryamong {O∆i ,li (Pi ,Zi )}, and differential operators only act on single(0, 0, l3) three point function.

I This is particularly useful when considering only the contribution ofsymmetric traceless operator O∆,J to the spinning 4 point function:

I Using the differential basis to express the 3 point functions, thespinning CPW can be related to scalar CPW as:

where Dn10,n20,n12

Left and Dn30,n40,n34

Right are the composite differentialoperators formed from Dij and Hij , i , j = 1, 2, 3, 4 plus shiftoperator Σa,b.


I We can also easily obtained the integral representation:[Simmons-Duffin 2012, HYC, Kuo, Kyono]

where τ (not twist ∆− J) include the effects of shift operator Σa,b. Wewill explain how identical expressions can be obtained holographically.


In AdSd+1/CFTd correspondence, CFT correlation functions are mappedto so-called Witten diagrams, the first non-trivial one is the 3pt vertex:

For our purpose, we can consider the following interaction vertex:

corresponding to holographic dual of scalar-scalar-tensor correlator.This yields a contribution to the 3pt correlation function as: [Costa et al 2014]

where W A is the bulk polarization vector and KA is its projector for indexcontractions.


Here the bulk-boundary propagator for massive spin-J particle is:

where τ = ∆ + J. Like in CFTd , AdS-isometries almost fix I0,0J :

where the overall factor b(∆1,∆2,∆, J) is the result of computation:

and the poles in ∆ capture double trace operators contributions, once weuse them to construct the four point functions.


Similarly, for general three spin interaction vertex, we have:[Sleight-Taronna 2016]

where the operators are given by:

Other combinations are equivalent up to integration by parts.

The integrated results can now be expressed in terms of linearcombination of box tensor structures (Diagonalization Problem):


Now for the 4 pt functions, we normally consider both contact andexchange diagrams:

They respectively yield the following contributions:

where Π∆,J(X , X ) is the spin J bulk to bulk propagator.


AdSd+1 analogue of splitting 4 point into product of 3 point functions isgiven by “split representation”: [Costa et al 2014]

We split the bulk-bulk propagator into products of bulk-boundary ones:

where aJ−k(ν), k = 0, 1, . . . , J are meromorphic functions in ν.


I We can now study the 3-point interaction vertices at X and X usingthe results earlier, and express them in terms of CFT structures:

I However when integrating with aJ−k(ν), the pre-factors from3-point functions, 4-point Witten diagram of spin-J exchange gives:

where the spectral function bl(ν) captures the physical spectrum:

Now we have both single trace and double trace contributions.


Q: What computes individual conformal blocks/partial waves?A: Geodesic Witten Diagrams. [Hijano et al 2015]

Quick Justification: What else? Instead of integrating over entireAdSd+1, we consider the trajectories which still preserve theAdS-isometries hence conformal symmetries.


More explicitly, we consider following geodesics:

and restrict the exchanged spinning particle to move along them.Such a 4-point geodesic Witten diagram gives the following:

This double integral along the geodesics can be shown such that:

by direct integration for low J exchange or by verifying that it satisfiesquadratic Casimir equation.


We can have an alternative proof by cutting 4-point geodesic Wittendiagram and consider the contributions from geodesic 3-point diagrams:[HYC-Kuo-Kyono]

Mathematically, this means we applied the following identity:

between the bulk to bulk and bulk to boundary propagators.


Generic 3-point interaction vertex along the geodesic, hC1...CJis divergent

free:

We have the following contribution, precisely the building block of CPW:

Fusing them back together using the kernal ν2

ν2+(h−∆)2 , we recovered

precisely the integral representation of scalar conformal block.


Scalar 4-point geodesic Witten diagrams can be regarded asdecomposition of Witten diagrams, by realizing their respective 3-pointbuilding elements only differ by ν-dependent overall factors, i .e.

Identical relation also exists for lower l spin, l = 0, . . . , J − 1 terms in thesummation, substitute back to P0 and ν integrations, we have:

We decompose the spin-J exchange Witten diagram into sum of geodesicWitten diagrams for single trace operator (Spin-J) and double traceoperator (Spin-l ≤ J). After non-trivial cancelation residues from thespurious poles.


For holographic reconstruction of spinning conformal blocks, we firstconsider geodesic 3-point functions for three primary operators with spins:

Arbitrary three point interaction vertex at X (λ) involving contractionswith metric and covariant derivatives yield sum of integrals like:

where τi = ∆i + li and Ql1,l2,l3 (Pi ,Zi ,X (λ)) is a transverse polynomial inZi of degree li .


I It is known that from earlier analysis: [Costa et al 2011]

the extra powers of e±λ can be absorbed as shift in ∆1,2, onlyscalar 3-point geodesic integral to do. Final result can be expressedin terms of linear combination of box tensor basis.

I Converting various box tensor basis into differential basis:

the fusion procedure then goes through identically as the scalargeodesic Witten diagram.

We have established that the spinning geodesic Witten diagrams generallyproduces linear combination of spinning conformal blocks for same O∆,J .


To provide dictionary between the bulk 3-point interaction vertices andthe spinning conformal blocks, adopt the parameterization of three spininteraction vertex earlier, with one important modification:

We have explicitly broken the cyclic properties among {O∆i ,li}, butnevertheless we still have the same number of independent vertices andbox tensor basis, as expected for linear combinations.

I For (l1 − l2 − 0), we have following vertices:

Direct computation then yields following contributions:


More general however, we have matrix diagonalization problem.

I For (1− 1− 2) there are five possible box tensor basis:

we label them in this ordering as {[Ir ]}, r = 1, 2, 3, 4, 5.

I There are also five possible vertices for 3-point geodesic WD:

their respective contributions are denoted as: {[Ir ]}, r = 1, 2, 3, 4, 5.

They are related by an invertible 5× 5 matrix [Ir ] = Trs [Is ], and for∆1 = ∆2,∆0 = ∆:


The Witten Diagrams also give us natural derivation for the Mellin partialamplitudes, or precisely the Mack polynomials(Slight change of notation for dynamical prefactors):

The kinematical integral Iν,J(Pi ) contributes to most of the scalar Mellinamplitudes after applying Symanzik star-formula:


I Stripping off the Γ-functions responsible for the double traceoperators and Pν,J(s, t) is given by: (Mack, HYC+Kuo+Kyono)

which is proportional to Mack polynomial Pν,J(s, t) up to spuriouspoles.

I The dynamical parts are encoded in the remaining pre-factors forthree point Witten functions and cutting kernel, and we can expressthem into ων,J(t) and the spectral function bJ(ν).


I For the derivation of spinning Mellin amplitude, we take similarstrategy and parameterize the spinning four point functions as:

The dynamical b(kL,R ,nL,R) depends on the specific basis forinteraction vertices, and determine the spectrum of double traceoperators.

I The kinematical I(nL,nR )ν,J (Pi ,Zi ) can be computed by generalizing

the Symanzik formula to include the polarization vectors {Zi}:(HYC+Kuo+Kyono)


This is a complicated long expression! But the salient features are:

I It now contains alternative basis of tensor structures Vij and Hij

which are linear combinations of Vi,jk and Hij .

I The parameters {δij} and non-negative integers {aij , bij} satisfy theconstraints:

I We can subsequently identify Mellin variables (s, t) in this case as:

I The generalized Mack polynomial Pν,J(s, t; aij , bij) does not containany poles in (s, t) variables.


Okay.....in case you are still wondering:


The proposal for the Mellin representation of the Spinning WittenDiagrams is:

We can show that the poles corresponding to the double trace operatorsin b(kL,R , lL,R) can be canceled by the zeros from denominators

Γ(δ12)Γ(δ34), hence M(nL,nR )ν,J (s, t; aij , bij) does not contain double trace

singularities in (s, t).


I Further constraining the vertices by imposing conservation/on-shellconditions?

I Generalization to other exchange channels with mixed symmetries?Fermionic geodesic Witten diagrams?

I Generalization to superconformal blocks? Compact directions?

I Useful for Mellin bootstrap for external operators with spins?


Spinning Witten Diagrams: From Geodesic to Mellin · 2020. 6. 24. · Spinning Witten Diagrams:...

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