A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics

TL;DR

This paper introduces the HarmonicSums computer algebra package, which manipulates harmonic sums, Euler-Zagier sums, and harmonic polylogarithms, aiding in the evaluation of Feynman integrals in particle physics.

Contribution

It presents a new algebraic framework and algorithms for manipulating harmonic sums and related quantities, including their differentiation and algebraic independence, with applications to Feynman integral evaluation.

Findings

Harmonic sums form a quasi-shuffle algebra.

New relations between harmonic sums are derived.

Algorithms support rewriting nested sums into harmonic sums.

Abstract

In this work we present the computer algebra package HarmonicSums and its theoretical background for the manipulation of harmonic sums and some related quantities as for example Euler-Zagier sums and harmonic polylogarithms. Harmonic sums and generalized harmonic sums emerge as special cases of so-called d'Alembertian solutions of recurrence relations. We show that harmonic sums form a quasi-shuffle algebra and describe a method how we can find algebraically independent harmonic sums. In addition, we define a differentiation on harmonic sums via an extended version of the Mellin transform. Along with that, new relations between harmonic sums will arise. Furthermore, we present an algorithm which rewrites certain types of nested sums into expressions in terms of harmonic sums. We illustrate by nontrivial examples how these algorithms in cooperation with the summation package Sigma support the evaluation of Feynman integrals.