Structural Relations of Harmonic Sums and Mellin Transforms up to Weight w = 5

TL;DR

This paper explores the structural relations of harmonic sums and Mellin transforms up to weight five, providing a reduced set of fundamental functions crucial for high-order calculations in QED and QCD.

Contribution

It extends the algebraic relations of harmonic sums beyond quasi-shuffle products, reducing the necessary functions to 15 basic meromorphic functions for complex calculations.

Findings

Reduced 69 harmonic sums to 30 using algebraic relations

Derived analytic representations for 15 fundamental functions

Provided asymptotic and recursion relations for these functions

Abstract

We derive the structural relations between the Mellin transforms of weighted Nielsen integrals emerging in the calculation of massless or massive single--scale quantities in QED and QCD, such as anomalous dimensions and Wilson coefficients, and other hard scattering cross sections depending on a single scale. The set of all multiple harmonic sums up to weight five cover the sums needed in the calculation of the 3--loop anomalous dimensions. The relations extend the set resulting from the quasi-shuffle product between harmonic sums studied earlier. Unlike the shuffle relations, they depend on the value of the quantities considered. Up to weight {\sf w = 5}, 242 nested harmonic sums contribute. In the present physical applications it is sufficient to consider the sub-set of harmonic sums not containing an index $i = -1$, which consists out of 69 sums. The algebraic relations reduce this set to 30 sums. Due to the structural relations a final reduction of the number of harmonic sums to {15} basic functions is obtained. These functions can be represented in terms of factorial series, supplemented by harmonic sums which are algebraically reducible. Complete analytic representations are given for these {15} meromorphic functions in the complex plane deriving their asymptotic- and recursion relations. A general outline is presented on the way nested harmonic sums and multiple zeta values emerge in higher order calculations of zero- and single scale quantities.