Application of the Meijer theorem in calculation of three-loop massive vacuum Feynman integrals and beyond

TL;DR

This paper introduces an analytical approach utilizing the Meijer theorem and Mellin-Barnes representation to compute complex three-loop massive Feynman integrals in arbitrary dimensions, with potential applications to other multi-loop integrals.

Contribution

It develops a novel analytical method based on the Meijer theorem for calculating multi-loop Feynman integrals, expanding the toolkit for quantum field theory computations.

Findings

Successfully applied to three-loop massive vacuum integrals

Demonstrated the method's applicability to other multi-loop integrals

Provides explicit formulas for integrals in arbitrary dimensions

Abstract

We present an analytical method to calculate the three-loop massive Feynman integral in arbitrary dimensions. The method is based on the Mellin-Barnes representation of the Feynman integral. The Meijer theorem and its corollary are used to perform the integration over the Gamma functions, exponential functions, and hypergeometric functions. We also discuss the application of the method in other multi-loop Feynman integrals.