Computer Algebra Algorithms for Special Functions in Particle Physics

TL;DR

This paper develops algorithms and software for manipulating special nested sums and integrals, crucial for high-order calculations in quantum field theories, and applies them to Feynman integrals.

Contribution

It introduces new algorithms and a computer algebra package for handling complex nested sums and integrals in particle physics calculations.

Findings

Algorithms for algebraic and structural relations of nested sums.

Implementation of the HarmonicSums package for symbolic computations.

Enhanced multivariate Almkvist-Zeilberger algorithm for Feynman integrals.

Abstract

This work deals with special nested objects arising in massive higher order perturbative calculations in renormalizable quantum field theories. On the one hand we work with nested sums such as harmonic sums and their generalizations (S-sums, cyclotomic harmonic sums, cyclotomic S-sums) and on the other hand we treat iterated integrals of the Poincar\'e and Chen-type, such as harmonic polylogarithms and their generalizations (multiple polylogarithms, cyclotomic harmonic polylogarithms). The iterated integrals are connected to the nested sums via (generalizations of) the Mellin-transformation and we show how this transformation can be computed. We derive algebraic and structural relations between the nested sums as well as relations between the values of the sums at infinity and connected to it the values of the iterated integrals evaluated at special constants. In addition we state algorithms to compute asymptotic expansions of these nested objects and we state an algorithm which rewrites certain types of nested sums into expressions in terms of cyclotomic S-sums. Moreover we summarize the main functionality of the computer algebra package HarmonicSums in which all these algorithms and transformations are implemented. Furthermore, we present application of and enhancements of the multivariate Almkvist-Zeilberger algorithm to certain types of Feynman integrals and the corresponding computer algebra package MultiIntegrate.