Harmonic Sums, Mellin Transforms and Integrals

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.

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

“…The graphs have been generated automatically with the diagram generator QGRAF [32]. All further symbolic manipulations have been performed in FORM [33,34], using the SUMMER package [35] [36,37] and the findings to Ref. [14].…”

“…Hence the next-to-next-to-next-to-leading order coefficient functions are now known, in massless perturbative QCD, for all structure functions for which precision measurements have been performed in fixedtarget DIS and/or at HERA, enabling improved analyses of such data at x > 0.01. Also the present calculation has been performed in Mellin-N space, obtaining an analytic formula in terms of harmonic sums [35] for all odd-N moments (as in Refs. [19] and [20] at first and second order) using the optical theorem and a dispersion relation in the Bjorken variable x.…”

Deep-inelastic structure functions: Reconstruction from Mellin moments

“…The contributing Feynman diagrams have been generated with QGRAF [30] and all recursion relations for the evaluation of the individual topologies have been programmed in FORM [31]. The nested sums which one encounters in this way are solved with the SUM-MER algorithm [16] in terms of the basis of harmonic sums. In addition, there is the possibility to perform checks at all stages of the calculation by means of the standard MINCER routine [32].…”

Multiloop calculations in HQET

“…This implementation is very convenient; unfortunately, it requires one to install an outdated version of GiNaC. The Algorithm A seems to be also implemented in FORM [20], but I could not understand how to use it. Using my REDUCE procedure or nestedsums [18], one can quickly find as many terms of expansion of this basis integral in ε as needed, in terms of multiple ζ-values.…”

“…Using my REDUCE procedure or nestedsums [18], one can quickly find as many terms of expansion of this basis integral in ε as needed, in terms of multiple ζ-values. They can be expressed, up to weight 9, via a minimum set of independent ζ-values, using the results of [22,20]. The two-loop diagram with a non-integer power of the middle line can also be expressed [21] via an Figure 1.…”