HPL, a Mathematica implementation of the harmonic polylogarithms

TL;DR

The paper presents a comprehensive Mathematica implementation of harmonic polylogarithms (HPLs) by Remiddi and Vermaseren, including product algebra, derivatives, singularity handling, series expansions, and numerical evaluation. It provides robust analytic continuation with user-controlled imaginary-part signs and introduces a rich set of options and utilities to manipulate HPLs efficiently. A minimal irreducible basis and detailed treatment of values at unity link HPLs to multiple zeta values, enabling reliable symbolic and numeric work in high-energy physics computations. The implementation is designed to integrate with broader Mathematica workflows and has been validated against existing implementations, with practical use cases such as HypExp.

Abstract

In this paper, we present an implementation of the harmonic polylogarithm of Remiddi and Vermaseren for Mathematica. It contains an implementation of the product algebra, the derivative properties, series expansion and numerical evaluation. The analytic continuation has been treated carefully, allowing the user to keep the control over the definition of the sign of the imaginary parts. Many options enables the user to adapt the behavior of the package to his specific problem.