Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications

TL;DR

This paper develops a Mellin space framework for conformal partial waves with arbitrary spin, enabling explicit calculation of crossing kernels and anomalous dimensions in conformal field theories.

Contribution

It introduces a factorisation property of spinning conformal partial waves in Mellin space, simplifying the derivation of crossing kernels and anomalous dimensions for operators of arbitrary spin.

Findings

Derived closed-form crossing kernels using hypergeometric functions.

Calculated anomalous dimensions of double-trace operators with arbitrary spin.

Provided explicit examples for various spins and twists.

Abstract

We study conformal partial waves (CPWs) in Mellin space with totally symmetric external operators of arbitrary integer spin. The exchanged spin is arbitrary, and includes mixed symmetry and (partially)-conserved representations. In a basis of CPWs recently introduced in arXiv:1702.08619, we find a remarkable factorisation of the external spin dependence in their Mellin representation. This property allows a relatively straightforward study of inversion formulae to extract OPE data from the Mellin representation of spinning 4pt correlators and in particular, to extract closed-form expressions for crossing kernels of spinning CPWs in terms of the hypergeometric function ${}_4F_3$. We consider numerous examples involving both arbitrary internal and external spins, and for both leading and sub-leading twist operators. As an application, working in general $d$ we extract new results for ${\cal O}\left(1/N\right)$ anomalous dimensions of double-trace operators induced by double-trace deformations constructed from single-trace operators of generic twist and integer spin. In particular, we extract the anomalous dimensions of double-trace operators $[\mathcal{O}_J\Phi]_{n,l}$ with ${\cal O}_J$ a single-trace operator of integer spin $J$.