New proofs for the two Barnes lemmas and an additional lemma

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 ∫dz involving a phase factor exp (±iπz).