Feynman Rules for Scalar Conformal Blocks

TL;DR

This paper completes the proof of Feynman rules for constructing scalar conformal blocks in any topology, dimension, and number of points, using Witten diagram interpretations and conformal cross ratios, with recursive proofs and examples.

Contribution

It provides a complete set of Feynman rules for scalar conformal blocks applicable to arbitrary topology, point number, and spacetime dimension, with a recursive proof and illustrative example.

Findings

Conformal blocks can be expressed as hypergeometric series in cross ratios.

The Feynman rules are validated through recursive proof techniques.

A nine-point example demonstrates the application of the rules.

Abstract

We complete the proof of "Feynman rules" for constructing $M$-point conformal blocks with external and internal scalars in any topology for arbitrary $M$ in any spacetime dimension by combining the rules for the blocks (based on their Witten diagram interpretation) with the rules for the construction of conformal cross ratios (based on OPE flow diagrams). The full set of Feynman rules leads to blocks as power series of the hypergeometric type in the conformal cross ratios. We then provide a proof by recursion of the Feynman rules which relies heavily on the first Barnes lemma and the decomposition of the topology of interest in comb-like structures. Finally, we provide a nine-point example to illustrate the rules.