Radial expansion for spinning conformal blocks

TL;DR

This paper introduces a new method to compute spinning conformal blocks using radial expansion, providing recursive relations and improved efficiency for certain cases, advancing the analytical tools in conformal field theory.

Contribution

It extends the radial expansion technique to spinning operators and derives recursion relations, enabling more efficient calculations of conformal blocks with spin.

Findings

Method applies to external operators with spin

Derived recursion relations for expansion coefficients

Enhanced efficiency for vector external operators

Abstract

This paper develops a method to compute any bosonic conformal block as a series expansion in the optimal radial coordinate introduced by Hogervorst and Rychkov. The method reduces to the known result when the external operators are all the same scalar operator, but it allows to compute conformal blocks for external operators with spin. Moreover, we explain how to write closed form recursion relations for the coefficients of the expansions. We study three examples of four point functions in detail: one vector and three scalars; two vectors and two scalars; two spin 2 tensors and two scalars. Finally, for the case of two external vectors, we also provide a more efficient way to generate the series expansion using the analytic structure of the blocks as a function of the scaling dimension of the exchanged operator.