Dimensional recurrence relations: an easy way to evaluate higher orders of expansion in $\epsilon$

TL;DR

This paper introduces a method using dimensional recurrence relations to analytically evaluate higher-order epsilon expansions of complex Feynman integrals, improving precision in quantum field theory calculations.

Contribution

The paper applies dimensional recurrence relations to obtain complete epsilon expansions of complex master integrals, including previously unknown constants and higher-order terms.

Findings

Analytical epsilon expansion of three-loop master integrals

Determination of three new terms in epsilon expansion of non-planar diagrams

Results involve only multiple zeta values at integer points

Abstract

Applications of a method recently suggested by one of the authors (R.L.) are presented. This method is based on the use of dimensional recurrence relations and analytic properties of Feynman integrals as functions of the parameter of dimensional regularization, $d$. The method was used to obtain analytical expressions for two missing constants in the $\epsilon$-expansion of the most complicated master integrals contributing to the three-loop massless quark and gluon form factors and thereby present the form factors in a completely analytic form. To illustrate its power we present, at transcendentality weight seven, the next order of the $\epsilon$-expansion of one of the corresponding most complicated master integrals. As a further application, we present three previously unknown terms of the expansion in $\epsilon$ of the three-loop non-planar massless propagator diagram. Only multiple $\zeta$ values at integer points are present in our result.