Mathematical Structure of Anomalous Dimensions and QCD Wilson Coefficients in Higher Order

TL;DR

Higher-order QED/QCD calculations generate large families of harmonic sums and related functions, making results unwieldy. The authors develop an algebraic framework based on shuffle algebra and Lyndon-word counting (via the 2nd Witt formula) to identify a minimal independent basis of harmonic sums up to depth 6 and extend counting to depth 10. They show structural and diagrammatic relations further reduce the required basic functions, propose a quadratic growth law for the basis size, and validate the approach by computing the 16th non-singlet 3-loop moment of $F_1(x,Q^2)$, agreeing with known results. The work provides practical methods to simplify and cross-check high-order QCD computations in Mellin space and for x-space translations, with broad applicability to massless gauge theories.

Abstract

The alternating and non-alternating harmonic sums and other algebraic objects of the same equivalence class are connected by algebraic relations which are induced by the product of these quantities and which depend on their index class rather than on their value. We show how to find a basis of the associated algebra. The length of the basis $l$ is found to be $\leq 1/d$, where $d$ is the depth of the sums considered and is given by the 2nd {\sc Witt} formula. It can be also determined counting the {\sc Lyndon} words of the respective index set. There are two further classes of relations: structural relations between {\sc Nielsen}--type integrals and relations due to the specific structure of {\sc Feynman} diagrams which lead to a considerable reduction of the set of basic functions. The relations derived can be used to simplify results of higher order calculations in QED and QCD. We also report on results calculating the 16th non--singlet moment of unpolarized structure functions at 3--loop order in the $\bar{\rm MS}$ scheme.