Analytic Continuation of the Harmonic Sums for the 3--Loop Anomalous Dimensions

TL;DR

The paper develops fast, highly accurate analytic continuations of eight new basic Mellin transforms needed for the 3--loop anomalous dimensions in QCD, building on the harmonic-sum framework. It introduces robust representations for transforms of the forms $M[f(x)/(1+x)](N)$ and $M[(f(x)/(1-x))_+](N)$ by combining minimax approximations with shifted moments and analytic near-singularity decompositions, providing explicit coefficient tables and verifiable accuracy. These representations enable precise, efficient numerical evaluations of massless Wilson coefficients and anomalous dimensions in Mellin space, with strong cross-checks against contour inversion. The work substantially extends the functional basis beyond 2--loops and provides practical tools (Fortran routines) for high-precision QCD evolution calculations.

Abstract

We present for numerical use the analytic continuations to complex arguments of those basic Mellin transforms, which build the harmonic sums contributing to the 3--loop anomalous dimensions. Eight new basic functions contribute in addition to the analytic continuations for the 2--loop massless Wilson coefficients calculated previously. The representations derived have a relative accuracy of better than $10^{-7}$ in the range $x ε[10^{-6},0.98]$.