Structural Relations between Harmonic Sums up to w=6

Abstract

or harmonic polylogarithms [5]. The simplification is, to some extent, due to the structure of Feynman parameter integrals which possess a Mellin symmetry. The respective expressions can be expressed by finite harmonic sums Sa1...an(N) for processes to 3–loop order [6, 7], which form the appropriate language. Within the light-cone expansion, or analogous formalisms for time-like processes, the argument of the harmonic sums are evenor odd integers, depending on the process. However, one may consider mathematical generalizations, continuing the argument analytically to rational, real and complex values N Q,R,C, respectively, [6,8]. In these extensions new relations between the harmonic sums are obtained, which lead to more compact representations. Since the hard-scattering cross sections usually have to be convoluted with non-perturbative parton densities, it is convenient to widely work in Mellin space using analytic representations, also for the solution of the evolution equations. This also applies to the treatment of heavy flavor contributions in the full phase space, for which concise semi-analytic representations were derived [9]. The final x−space results are obtained by a single numerical Mellin–inversion performed by a contour integral around the singularities of the problem. In this note we give a summary on the algebraic and structural relations for finite harmonic sums, occurring in hard scattering processes. As an example we consider the virtual and soft QED corrections to Bhabha-scattering to O(α) in the on-mass-shell scheme [10] to show that also this process fits to the general basis-representation being derived for various other two– and three–loop QCD processes.