Gauss Relations in Feynman Integrals

TL;DR

This paper introduces a systematic method to derive Gauss relations among hypergeometric functions representing Feynman integrals, enabling analytic continuation and Laurent series expansion around four dimensions.

Contribution

It presents a general approach to obtain Gauss relations for hypergeometric functions in Feynman integrals, facilitating their analytic continuation and series expansion.

Findings

Derived Gauss relations for hypergeometric functions in Feynman integrals.

Applied method to 1-loop self-energy and 2-loop massless triangle diagrams.

Enabled Laurent series expansion around D=4 using Gauss adjacent relations.

Abstract

Embedding Feynman integrals in Grassmannians, we can write Feynman integrals as some finite linear combinations of generalized hypergeometric functions. In this paper we present a general method to obtain Gauss relations among those generalized hypergeometric functions. The hypergeometric expressions of Feynman integral are continued from a connected component to another by the inverse Gauss relations, then continued to the whole domain of definition by the Gauss-Kummer relations. The Laurent series of the Feynman integral around the space-time dimension $D=4$ is obtained by the Gauss adjacent relations where the coefficient of the term with power of $D-4$ is given as the linear combinations of hypergeometric functions with integer parameters. As examples, we illustrate how to obtain the expressions for the Feynman integrals of the 1-loop self-energy and a 2-loop massless triangle diagram in the domains of definition.