Tuning FORM with large calculations

Some recent additions to FORM are discussed. In particular large file support and the tablebases are presented.Comment: 5 pages, LaTeX. Talk given at RADCOR02, Kloster Ban

- XSummer - Transcendental Functions and Symbolic Summation in Form

Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory. Of particular interest have been the so-called nested sums,where the harmonic sums and their generalizations appear as building blocks, originating for example from the expansion of generalized hypergeometric functions around integer values of the parameters. In this Letter we discuss the implementation of several algorithms to solve these sums by algebraic means, using the computer algebra system Form.Comment: 21 pages, 1 figure, Late

Harmonic sums, Mellin transforms and Integrals

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.Comment: 31 pages LaTeX, for programs, see http://norma.nikhef.nl/~t68/summe

Mathematics for structure functions

We show some of the mathematics that is being developed for the computation of deep inelastic structure functions to three loops. These include harmonic sums, harmonic polylogarithms and a class of difference equations that can be solved with the use of harmonic sums.Comment: 6 pages LaTeX, uses axodraw.sty and npb.sty (included

The analytic value of the sunrise self-mass with two equal masses and the external invariant equal to the third squared mass

We consider the two-loop self-mass sunrise amplitude with two equal masses $M$ and the external invariant equal to the square of the third mass $m$ in the usual $d$-continuous dimensional regularization. We write a second order differential equation for the amplitude in $x=m/M$ and show as solve it in close analytic form. As a result, all the coefficients of the Laurent expansion in $(d-4)$ of the amplitude are expressed in terms of harmonic polylogarithms of argument $x$ and increasing weight. As a by product, we give the explicit analytic expressions of the value of the amplitude at $x=1$, corresponding to the on-mass-shell sunrise amplitude in the equal mass case, up to the $(d-4)^5$ term included.Comment: 11 pages, 2 figures. Added Eq. (5.20) and reference [4

The reaction e+e- --> hh recomputed

We notice that the existing literature about the reaction e+e- --> hh suffers from a mistake in the relative sign between the t-channel and u-channel graphs. Correcting this lowers the crosssections by about an order of magnitude.Comment: 7 pages, 5 figure

Hypergeometric representation of a four-loop vacuum bubble

In this article, we present a new analytic result for a certain single-mass-scale four-loop vacuum (bubble) integral. We also discuss its systematic \e-expansion in d=4-2\e as well as d=3-2\e dimensions, the coefficients of which are expressible in terms of harmonic sums at infinity.Comment: 5 pages, to appear in the proceedings of the conference "Loops and Legs", Eisenach, 200

The diamond rule for multi-loop Feynman diagrams

An important aspect of improving perturbative predictions in high energy physics is efficiently reducing dimensionally regularised Feynman integrals through integration by parts (IBP) relations. The well-known triangle rule has been used to achieve simple reduction schemes. In this work we introduce an extensible, multi-loop version of the triangle rule, which we refer to as the diamond rule. Such a structure appears frequently in higher-loop calculations. We derive an explicit solution for the recursion, which prevents spurious poles in intermediate steps of the computations. Applications for massless propagator type diagrams at three, four, and five loops are discussed