On the Method of Brackets

TL;DR

This paper compares the Method of Brackets and Mellin-Barnes techniques for evaluating integrals, highlighting the advantages of MB in handling divergent series and resonant cases, and clarifies the connection between the two methods.

Contribution

It provides a detailed comparison of MoB and MB, demonstrating MB's superiority in convergence analysis and resonant cases, and explains how Rule 5 of MoB arises from MB's residue theorem.

Findings

MB uses conic hulls for series selection, avoiding convergence issues.

MB effectively evaluates resonant (logarithmic) integrals where MoB fails.

Rule 5 of MoB naturally follows from MB's residue theorem.

Abstract

The Method of Brackets (MoB) is a technique used to compute definite integrals, that has its origin in the negative dimensional integration method. It was originally proposed for the evaluation of Feynman integrals for which, when applicable, it gives the results in terms of combinations of (multiple) series. We focus here on some of the limitations of MoB and address them by studying the Mellin-Barnes (MB) representation technique. There has been significant process recently in the study of the latter due to the development of a new computational approach based on conic hulls (see Phys. Rev. Lett. 127, 151601 (2021)). The comparison between the two methods helps to understand the limitations of the MoB, in particular when termwise divergent series appear. As a consequence, the MB technique is found to be superior over MoB for two major reasons: 1. the selection of the sets of series that form a series representation for a given integral follows, in the MB approach, from specific intersections of conic hulls, which, in contrast to MoB, does not need any convergence analysis of the involved series, and 2. MB can be used to evaluate resonant (i.e. logarithmic) cases where MoB fails due to the appearance of termwise divergent series. Furthermore, we show that the recently added Rule 5 of MoB naturally emerges as a consequence of the residue theorem in the context of MB.