A simple way to reduce the number of contours in the multi-fold Mellin-Barnes integrals

TL;DR

This paper introduces a novel method to reduce the number of contours in multi-fold Mellin-Barnes integrals for Feynman diagrams, simplifying calculations without relying on Barnes lemma.

Contribution

It presents a new approach using analytical regularization and the integral Cauchy formula to reduce five-fold integrals to two-fold, applicable to arbitrary Feynman diagrams.

Findings

Successfully reduces five-fold to two-fold Mellin-Barnes integrals.

Connects the reduction method to knot theory and quantum computing.

Provides a practical strategy for simplifying complex Feynman integrals.

Abstract

Mellin-Barnes integral representation of one-loop off-shell box massless diagram is five-fold by construction. On the other hand, it is known from the year 1992 that it may be reduced to certain two-fold Mellin-Barnes integral. We propose a way to reduce the number of the Mellin-Barnes integration contours from five to two by using the Mellin-Barnes integral representation only in combination with basic methods of mathematical analysis such as analytical regularization. We do not use any Barnes lemma to prove the reduction but we use the integral Cauchy formula instead. We recover first the well-known two-fold Mellin-Barnes representation for the one-loop triangle massless diagram and then show how the five-fold Mellin-Barnes integral representation of one-loop box diagram with all the indices 1 in four spacetime dimensions may be reduced to the two-fold Mellin-Barnes representation for one-loop triangle diagram. Singular integrals over Feynman parameters appear in the integrand of the five-fold Mellin-Barnes integral representation at the intermediate step. Such integrals should be treated as distributions with respect to certain linear combinations of the initial Mellin-Barnes integration variables in the Mellin-Barnes integrands. These distributions may be integrated out with a finite number of residues in the limit of removing the analytical regularization. We explain how to apply this strategy to an arbitrary Feynman diagram in order to reduce the number of Mellin-Barnes integration contours. On the practical side, we analyze connections between the obtained results and the knot theory, Trotter integrals, quantum computing.