Harmonic sums and Mellin transforms up to two-loop order

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 timelike 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. ͓S0556-2821͑99͒04411-2͔

Third-order QCD corrections to the charged-current structure function

“…As discussed above, our calculation via the optical theorem and a dispersion relation directly determines the coefficient function (3.1) for all odd-integer moments N in terms of harmonic sums [35,39,40]. From these functions the x-space expressions can be reconstructed algebraically [20,43] in terms of harmonic polylogarithms [41][42][43].…”

“…(3.6) and (3.7) can be readily transformed to Mellin space at complex values of N (using, e.g., the appendix of Ref. [40] for the moments of ln x ln 2 (1 − x) etc) for use with N-space programs, such as QCD-PEGASUS [48], for the evolution of parton densities and structure functions.…”

Scheme-invariant NNLO evolution for unpolarized DIS structure functions

“…Here γ N mn denotes the unpolarized anomalous dimensions which are related to the evolution kernels in x−space by (10) and C N I,m are the Mellin transforms of the Wilson coefficients…”

“…We therefore perform the evolution in Mellin space. The result, computed for integer N is then analytically continued to arbitrary complex N using representations of harmonic sums with arbitrary precision [10] and the x−space result is obtained through a single numerical contour integral.…”

Deep-inelastic structure functions: Reconstruction from Mellin moments

“…functions in terms of harmonic sums [16,17] and the subsequent analytical reconstruction based on an inverse Mellin transformation to harmonic polylogarithms [18] in x-space. This procedure as detailed in [19,20] has recently been used [19] to check the original calculation of the two-loop coefficient functions [21][22][23][24], which was performed with conventional techniques.…”