IBIS: Inverse BInomial sum Solver

TL;DR

The paper introduces IBIS, a FORM program that efficiently computes inverse binomial sums in higher-loop Feynman integrals, expanding the class of sums that can be analytically evaluated using $S$-sums.

Contribution

It extends the calculation of sums involving inverse binomials, providing new recursion relations and a computational tool for higher-loop quantum field theory calculations.

Findings

Inverse binomial sums can be expressed in terms of $S$-sums.

IBIS performs these calculations rapidly, handling sums up to weight 6 in less than a second.

The method facilitates the study of nested sums in complex Feynman integrals.

Abstract

In higher-loop calculations, Mellin-Barnes representations are used to simplify the denominators encountered in Feynman parameter integrals. The contour integral of these representations yield sums over residues. We extend the classes of such sums that can be calculated, to include those involving inverse binomials. Our results are expressed in terms of so-called $S$-sums, where the dependence on the upper limit of the sum is analytic. This is accomplished by deriving several new recursion relations, obtained from telescoping series and repeated synchronization. We make our results available through IBIS ("Inverse BInomial sum Solver"), a FORM program to perform such inverse binomial sums. It expresses them in terms of $S$-sums, which can be handled by XSUMMER. To illustrate the efficiency of our code: sums up to weight 6 can be carried out in less than a second. We show in an example how inverse binomial sums arise, though in many instances these are nested with binomial sums, beyond the cases studied here. Our work thus provides a starting point for studying such sums, that will open up new avenues in higher-loop calculations.