Towards Feynman rules for conformal blocks

TL;DR

This paper proposes a conjectural set of Feynman rules for constructing conformal blocks in any channel and dimension, supported by proofs for known cases and new topologies using holographic methods.

Contribution

It introduces a simple, explicit set of Feynman rules for conformal blocks applicable to any number of points and dimensions, with proofs for various cases.

Findings

Rules reproduce known four-, five-, six-point blocks

Extended to seven-point and even-point blocks in new topologies

Proofs rely on holographic Mellin amplitude methods

Abstract

We conjecture a simple set of "Feynman rules" for constructing $n$-point global conformal blocks in any channel in $d$ spacetime dimensions, for external and exchanged scalar operators for arbitrary $n$ and $d$. The vertex factors are given in terms of Lauricella hypergeometric functions of one, two or three variables, and the Feynman rules furnish an explicit power-series expansion in powers of cross-ratios. These rules are conjectured based on previously known results in the literature, which include four-, five- and six-point examples as well as the $n$-point comb channel blocks. We prove these rules for all previously known cases, as well as for a seven-point block in a new topology and the even-point blocks in the "OPE channel." The proof relies on holographic methods, notably the Feynman rules for Mellin amplitudes of tree-level AdS diagrams in a scalar effective field theory, and is easily applicable to any particular choice of a conformal block.