Harmonic Polylogarithms

TL;DR

The paper introduces harmonic polylogarithms (hpl's) as a recursive, Nielsen-generalizing class of functions tailored for multi-loop, multi-scale calculations, and establishes their rich algebraic structure, including product and transformation identities. It connects hpl's to harmonic sums via their series and Mellin transforms, and develops systematic methods for reductions, special-value evaluations, and numerical implementations (notably in FORM). A central contribution is a detailed framework for handling identities, related-argument transformations, and the inverse Mellin transform, enabling efficient representation of high-weight results in both x- and Mellin-space. The work provides practical algorithms for extracting logarithmic terms, computing special values (e.g., at x=1/2), and evaluating Mellin transforms, which are essential for perturbative calculations in quantum field theory. Overall, it offers a comprehensive toolkit for manipulating and numerically evaluating a broad class of polylogarithmic functions beyond Nielsen polylogarithms, with direct relevance to high-precision loop computations.

Abstract

The harmonic polylogarithms (hpl's) are introduced. They are a generalization of Nielsen's polylogarithms, satisfying a product algebra (the product of two hpl's is in turn a combination of hpl's) and forming a set closed under the transformation of the arguments x=1/z and x=(1-t)/(1+t). The coefficients of their expansions and their Mellin transforms are harmonic sums.