$O(α_s^2)$ Timelike Wilson Coefficients for Parton-Fragmentation Functions in Mellin Space

TL;DR

This work derives the $O(\alpha_s^2)$ timelike Wilson coefficients for parton-fragmentation functions in Mellin space, expressing them through a compact set of nested harmonic sums with maximal weight 4. By exploiting algebraic and structural relations among harmonic sums and their analytic continuations to complex Mellin variable $N$, the authors show that only three basic sums (plus derivatives) are needed beyond single sums, greatly simplifying the functional basis compared to $x$-space representations. The Mellin-space formulation enables analytic solutions to evolution equations and fast, precise numerical implementations, facilitating the use of fragmentation data in precision QCD analyses. The appendices provide explicit transforms for unpolarized and polarized cases, and cross-channel consistency with space-like results is highlighted via physical evolution kernels.

Abstract

We calculate the Mellin moments of the $O(α_s^2)$ coefficient functions for the unpolarized and polarized fragmentation functions. They can be expressed in terms of multiple finite harmonic sums of maximal weight {\sf w = 4}. Using algebraic and structural relations between harmonic sums one finds that besides the single harmonic sums only three basic sums and their derivatives w.r.t. the summation index contribute. The Mellin moments are analytically continued to complex values of the Mellin variable. This representation significantly reduces the large complexity being present in $x$--space calculations and allow very compact and fast numerical implementations.