Algebraic Relations Between Harmonic Sums and Associated Quantities

TL;DR

This work develops a comprehensive, index-structure–driven framework for the algebraic relations among finite harmonic sums up to depth 6, showing that all relations arise from shuffle-product structures and permutations of the index-set. By connecting these sums to harmonic polylogarithms and employing Lyndon-word counting via the Witt-Radford framework, the author derives explicit representations of all sums in terms of a minimal basis, with the basis size tied to the number of Lyndon words. The results significantly reduce the number of independent objects needed to express high-order Mellin-space quantities in massless QED/QCD and extend to related nested-sum objects, providing analytic forms and tables up to depth 10. The methodology offers a systematic route to simplify complex multi-loop calculations by pre-eliminating redundant sums using algebraic relations determined solely by index structure. Overall, the paper establishes a powerful combinatorial and algebraic toolkit for manipulating harmonic sums and their applications in perturbative quantum field theory.

Abstract

We derive the algebraic relations of alternating and non-alternating finite harmonic sums up to the sums of depth~6. All relations for the sums up to weight~6 are given in explicit form. These relations depend on the structure of the index sets of the harmonic sums only, but not on their value. They are therefore valid for all other mathematical objects which obey the same multiplication relation or can be obtained as a special case thereof, as the harmonic polylogarithms. We verify that the number of independent elements for a given index set can be determined by counting the Lyndon words which are associated to this set. The algebraic relations between the finite harmonic sums can be used to reduce the high complexity of the expressions for the Mellin moments of the Wilson coefficients and splitting functions significantly for massless field theories as QED and QCD up to three loop and higher orders in the coupling constant and are also of importance for processes depending on more scales. The ratio of the number of independent sums thus obtained to the number of all sums for a given index set is found to be $\leq 1/d$ with $d$ the depth of the sum independently of the weight. The corresponding counting relations are given in analytic form for all classes of harmonic sums to arbitrary depth and are tabulated up to depth $d=10$.