Nested (inverse) binomial sums and new iterated integrals for massive Feynman diagrams

TL;DR

This paper develops methods to analyze nested binomial sums and their related iterated integrals, providing algorithms and basis choices to facilitate calculations in massive Feynman diagram evaluations.

Contribution

It introduces new algorithms and basis constructions for converting nested sums with binomial coefficients into iterated integrals involving radicals, enhancing computational techniques for Feynman diagrams.

Findings

Established a suitable basis for integrals with radicals

Developed algorithms for sum-integral conversion using Mellin transforms

Derived rewrite rules and identified patterns in the conversion process

Abstract

Nested sums containing binomial coefficients occur in the computation of massive operator matrix elements. Their associated iterated integrals lead to alphabets including radicals, for which we determined a suitable basis. We discuss algorithms for converting between sum and integral representations, mainly relying on the Mellin transform. To aid the conversion we worked out dedicated rewrite rules, based on which also some general patterns emerging in the process can be obtained.