Hypergeometric Functions and Feynman Diagrams

TL;DR

This paper explores the connection between Feynman diagrams and hypergeometric functions, focusing on epsilon-expansions and their applications to complex diagrams like the massless pentagon, revealing limitations of polylogarithmic expressibility.

Contribution

It introduces new methods for constructing epsilon-expansions of hypergeometric functions, including the Appell function F_3, and analyzes their application to Feynman diagrams, highlighting cases not expressible by polylogarithms.

Findings

Epsilon-expansion of Appell function F_3 constructed around rational parameters.

Massless pentagon diagram in 3-2ε dimensions not expressible via multiple polylogarithms.

Identification of Puiseux-type solutions involving hypergeometric functions of three variables.

Abstract

The relationship between Feynman diagrams and hypergeometric functions is discussed. Special attention is devoted to existing techniques for the construction of the $\epsilon$-expansion. As an example, we present a detailed discussion of the construction of the epsilon-expansion of the Appell function $F_3$ around rational values of parameters via an iterative solution of differential equations. As a by-product, we have found that the one-loop massless pentagon diagram in dimension $d=3-2\epsilon$ is not expressible in terms of multiple polylogarithms. Another interesting example is the Puiseux-type solution involving a differential operator generated by a hypergeometric function of three variables. The holonomic properties of the $F_N$ hypergeometric functions are briefly discussed.