Analytic Continuation of Mellin Transforms up to two-loop Order

TL;DR

Blümlein delivers a semi-analytic framework for the analytic continuation of Mellin transforms of 25 Nielsen-based basic functions up to two-loop order in massless QED/QCD, enabling accurate complex-N representations and Mellin-space inversions. By expressing integer-N transforms through finite harmonic sums and deriving complex-N formulas, the work supports efficient construction of two-loop splitting, coefficient, and hard-scattering functions. The ANCONT Fortran77 code embodies these representations, providing high-precision, test-validated tools for Mellin inversions and cross-section calculations in both space- and time-like processes. Overall, this work furnishes a practical, rigorous pathway to assemble all two-loop massless gauge-theory observables in Mellin space with robust numerical accuracy.

Abstract

The analytic continuation of the Mellin transforms to complex values of N for the basic functions $g_i(x)$ 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 are evaluated. These Mellin transforms provide the analytic continuations of all finite harmonic sums up to the level of the threefold sums of transcendentality four, where the basis-set ${g_i(x)}$ consists of products of {\sc Nielsen}-integrals up to transcendentality four. The computer code {\tt ANCONT} is provided.