Structural Relations between Nested Harmonic Sums

TL;DR

This paper analyzes the structural relations among nested harmonic sums at weight 6, crucial for describing physical quantities in gauge theories up to three loops, and identifies universal functions for their representation.

Contribution

It introduces a systematic approach to relate nested harmonic sums and identifies a minimal set of basic functions for three-loop calculations in QCD.

Findings

35 basic functions describe 3-loop Wilson coefficients

15 functions suffice for 3-loop anomalous dimensions

Complex analysis of these functions is derived

Abstract

We describe the structural relations between nested harmonic sums emerging in the description of physical single scale quantities up to the 3--loop level in renormalizable gauge field theories. These are weight {\sf w=6} harmonic sums. We identify universal basic functions which allow to describe a large class of physical quantities and derive their complex analysis. For the 3--loop QCD Wilson coefficients 35 basic functions are required, whereas a subset of 15 describes the 3--loop anomalous dimensions.