Harmonic sums, Mellin transforms and Integrals

TL;DR

The paper develops a comprehensive, FORM-based framework (SUMMER) for symbolic manipulation of nested harmonic sums, Mellin transforms, and inverse Mellin transforms as they appear in Feynman-diagram calculations. It introduces precise notations, synchronization techniques, and basis-conversion algorithms to handle complex sums, including divergent contributions regularized via S_1(∞). It also provides methods to evaluate sums at infinity (Euler-Zagier-type) and to construct inverse Mellin transforms from weight-bym-weight constructions, enabling analytic evaluation of broad classes of integrals and DIS moments. The results are complemented by extensive algorithmic details, practical implementations, and tables up to weight seven, with applications to two- and potentially three-loop diagrams and broader QCD computations.

Abstract

This paper describes algorithms to deal with nested symbolic sums over combinations of harmonic series, binomial coefficients and denominators. In addition it treats Mellin transforms and the inverse Mellin transformation for functions that are encountered in Feynman diagram calculations. Together with results for the values of the higher harmonic series at infinity the presented algorithms can be used for the symbolic evaluation of whole classes of integrals that were thus far intractable. Also many of the sums that had to be evaluated seem to involve new results. Most of the algorithms have been programmed in the language of FORM. The resulting set of procedures is called SUMMER.