Recursion relations for 5-point conformal blocks

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.

Automated summary

In this study, we focus on 5-point functions in conformal field theories with dimensions greater than 2. By utilizing weight-shifting operators, we are able to derive recursion relations that simplify the computation of arbitrary conformal blocks in 5-point functions involving scalar operators, reducing them to a linear combination of blocks with scalar exchanges. Additionally, we establish recursion relations for the conformal blocks with spin 1 or 2 in the context of 5-point functions, allowing for the formulation of positivity constraints related to the expectation value of the energy operator in bilocal states created by two scalars.