Recursion relations for 5-point conformal blocks

TL;DR

This paper develops recursion relations for 5-point conformal blocks in higher-dimensional conformal field theories, enabling efficient computation and analysis of these blocks, including cases with spinning operators.

Contribution

It introduces new recursion relations for 5-point conformal blocks using weight-shifting operators, extending to cases with external operators of spin 1 or 2, and applies these to positivity constraints.

Findings

Derived recursion relations for scalar 5-point blocks

Extended recursion relations to spinning operators

Formulated positivity constraints using 5-point functions

Abstract

We consider 5-point functions in conformal field theories in d > 2 dimensions. Using weight-shifting operators, we derive recursion relations which allow for the computation of arbitrary conformal blocks appearing in 5-point functions of scalar operators, reducing them to a linear combination of blocks with scalars exchanged. We additionally derive recursion relations for the conformal blocks which appear when one of the external operators in the 5-point function has spin 1 or 2. Our results allow us to formulate positivity constraints using 5-point functions which describe the expectation value of the energy operator in bilocal states created by two scalars.