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 time-like 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.

44 pages Latex, contract number added

References in corpus (3)

• The Evolution of Unpolarized Singlet Structure Functions at Small x
• O(α_s^2) contributions to the fragmentation function g_1(x,Q^2) in polarized e^+e^- annihilation
• Relations among polarized and unpolarized splitting functions beyond leading order

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