New proofs for the two Barnes lemmas and an additional lemma

TL;DR

This paper introduces new, elegant proofs for the Barnes lemmas used in evaluating Feynman integrals, avoiding hypergeometric functions, and presents an additional lemma involving phase factors in Mellin-Barnes integrals.

Contribution

It provides novel proofs of the Barnes lemmas using basic MB identities, making the results more accessible for educational purposes and introducing a new lemma involving phase factors.

Findings

New proofs rely solely on MB identities, avoiding series summations.

The paper introduces an additional lemma involving phase factors in MB integrals.

Proofs are simplified and more accessible for students of quantum field theory.

Abstract

Mellin-Barnes (MB) representations have become a widely used tool for the evaluation of Feynman loop integrals appearing in perturbative calculations of quantum field theory. Some of the MB integrals may be solved analytically in closed form with the help of the two Barnes lemmas which have been known in mathematics already for one century. The original proofs of these lemmas solve the integrals by taking infinite series of residues and summing these up via hypergeometric functions. This paper presents new, elegant proofs for the Barnes lemmas which only rely on the well-known basic identity of MB representations, avoiding any series summations. They are particularly useful for presenting and proving the Barnes lemmas to students of quantum field theory without requiring knowledge on hypergeometric functions. The paper also introduces and proves an additional lemma for a MB integral \int dz involving a phase factor exp(+-i pi z).