Harmonic Sums and Mellin Transforms up to two-loop Order

TL;DR

The paper demonstrates that all Nielsen-function–driven Mellin transforms arising in massless QED/QCD up to two loops can be represented as linear combinations of finite harmonic sums up to depth four. It provides explicit linear representations and a rich set of algebraic relations that reduce the required basis to about 20 core Mellin transforms, with fourfold sums appearing only in simple, reducible forms at this order. The results enable compact, extensible Mellin-space expressions for splitting and coefficient functions and are validated against known two-loop results. An extensive appendix tabulates the necessary Mellin transforms, facilitating high-precision, complex-N analyses in perturbative QCD/QED.

Abstract

A systematic study is performed on the finite harmonic sums up to level four. These sums form the general basis for the Mellin transforms of all individual functions $f_i(x)$ of the momentum fraction $x$ emerging in the quantities of massless QED and QCD up to two--loop order, as the unpolarized and polarized splitting functions, coefficient functions, and hard scattering cross sections for space and time-like momentum transfer. The finite harmonic sums are calculated explicitly in the linear representation. Algebraic relations connecting these sums are derived to obtain representations based on a reduced set of basic functions. The Mellin transforms of all the corresponding Nielsen functions are calculated.