A Feynman Integral and its Recurrences and Associators

TL;DR

This paper derives explicit epsilon-expansions for hypergeometric functions related to Feynman integrals using recurrence relations and associators, providing a systematic approach for all-order expansions in quantum field theory calculations.

Contribution

It introduces a method combining differential equations and recurrence relations with non-commutative coefficients to compute epsilon-expansions of hypergeometric functions and Feynman integrals.

Findings

Explicit epsilon-expansions for hypergeometric functions are obtained.

Regularized zeta series and associators are computed for these functions.

All-order epsilon-expansions for Feynman integrals are demonstrated.

Abstract

We determine closed and compact expressions for the epsilon-expansion of certain Gaussian hypergeometric functions expanded around half-integer values by explicitly solving for their recurrence relations. This epsilon-expansion is identified with the normalized solution of the underlying Fuchs system of four regular singular points. We compute its regularized zeta series (giving rise to two independent associators) whose ratio gives the epsilon-expansion at a specific value. Furthermore, we use the well known one-loop massive bubble integral as an example to demonstrate how to obtain all-order epsilon-expansions for Feynman integrals and how to construct representations for Feynman integrals in terms of generalized hypergeometric functions. We use the method of differential equations in combination with the recently established general solution for recurrence relations with non-commutative coefficients.