Mathematics for structure functions

TL;DR

The paper addresses the challenge of computing three-loop deep inelastic structure functions by developing a Mellin-space framework built on harmonic sums $S_{m_1,\dots,m_k}(N)$, harmonic polylogarithms $H(\vec m_w;x)$, and their Mellin/ inverse-Mellin transforms, together with a difference-equation solving paradigm. It details how HPs and HSs are connected across weight via Mellin transforms, and presents a practical algorithm to invert HP expansions back to HS representations, including even/odd moment handling. A general, constructive method for solving high-order difference equations that relate complex diagrams to simpler building blocks is described, with boundary data from Mincer used to fix constants. The work constitutes a scalable mathematical toolkit aimed at producing three-loop anomalous dimensions and coefficient functions for QCD structure functions, with substantial progress and clear computational challenges outlined for completing the hierarchy of diagrams.

Abstract

We show some of the mathematics that is being developed for the computation of deep inelastic structure functions to three loops. These include harmonic sums, harmonic polylogarithms and a class of difference equations that can be solved with the use of harmonic sums.