PolyLogTools - Polylogs for the masses

TL;DR

This work surveys the mathematical structure of multiple polylogarithms (MPLs), their Hopf-algebra coproduct, and the symbol map, and discusses single-valued MPLs and fibration-basis methods. It introduces PolyLogTools, a Mathematica package implementing these structures to manipulate MPLs, compute coproducts and symbols, and facilitate Feynman-integral calculations at low loop orders. The paper details algorithmic tools for shuffle/stuffle algebras, special-value reductions, integrals and series, and numerical evaluation, and demonstrates these with applications to high-energy physics and a representative one-loop box calculation. Overall, PolyLogTools provides a coherent, tested framework for exploring MPL identities, basis transformations, and Feynman-parameter integrals, supporting both practical computations and formal investigations of MPL structures.

Abstract

We review recent developments in the study of multiple polylogarithms, including the Hopf algebra of the multiple polylogarithms and the symbol map, as well as the construction of single valued multiple polylogarithms and discuss an algorithm for finding fibration bases. We document how these algorithms are implemented in the Mathematica package PolyLogTools and show how it can be used to study the coproduct structure of polylogarithmic expressions and how to compute iterated parametric integrals over polylogarithmic expressions that show up in Feynman integal computations at low loop orders.

Figures (1)

• Figure 1: The four-mass box integral.