Structural Relations of Harmonic Sums and Mellin Transforms at Weight w=6

TL;DR

This paper establishes structural relations between harmonic sums and Mellin transforms at weight 6, reducing the complexity of functions needed for high-order calculations in QED and QCD.

Contribution

It introduces a comprehensive set of structural relations at weight 6, reducing the number of basic functions for physical calculations and providing an algorithm for their analytic representation.

Findings

Reduced harmonic sums from 486 to 20 basic functions at weight 6.

Extended the set of basic functions from weight 5 to 6.

Provided an algorithm for analytic representation in the complex plane.

Abstract

We derive the structural relations between nested harmonic sums and the corresponding Mellin transforms of Nielsen integrals and harmonic polylogarithms at weight {\sf w = 6}. They emerge in the calculations of massless single--scale quantities in QED and QCD, such as anomalous dimensions and Wilson coefficients, to 3-- and 4--loop order. We consider the set of the multiple harmonic sums at weight six without index $\{-1\}$. This restriction is sufficient for all known physical cases. The structural relations supplement the algebraic relations, due to the shuffle product between harmonic sums, studied earlier. The original amount of 486 possible harmonic sums contributing at weight {\sf w = 6} reduces to 99 sums with no index $\{-1\}$. Algebraic and structural relations lead to a further reduction to 20 basic functions. These functions supplement the set of 15 basic functions up to weight {\sf w = 5} derived formerly. We line out an algorithm to obtain the analytic representation of the basic sums in the complex plane.