On the Mellin Transform of the Coefficient Functions of $F_L(x,Q^2)$

TL;DR

The paper addresses the problem of obtaining Mellin-space representations for the NLO Wilson coefficient functions of the longitudinal structure function $F_L(x,Q^2)$ and their analytic continuation. It develops explicit Mellin transforms of the $x$-space coefficients, leveraging harmonic sums and polygamma functions and introducing simplifications to polylogarithmic terms, along with fast-converging representations for numerically stable evaluation. The authors provide tables of basis-function transforms, discuss the handling of alternating sums, and present practical schemes for accurate computation of $M[F_L](N)$ in Mellin space, enabling direct NLO calculations without repeated inverse transforms. This work enhances the efficiency and consistency of QCD scaling-violation analyses by enabling Mellin-space treatment of $F_L$.

Abstract

The Mellin-transforms of the next-to-leading order Wilson coefficients of the longitudinal structure function are evaluated.