Schouten identities for Feynman graph amplitudes; the Master Integrals for the two-loop massive sunrise graph

A new class of identities for Feynman graph amplitudes, dubbed Schouten identities, valid at fixed integer value of the dimension d is proposed. The identities are then used in the case of the two loop sunrise graph with arbitrary masses for recovering the second order differential equation for the scalar amplitude in d=2 dimensions, as well as a chained sets of equations for all the coefficients of the expansions in (d-2). The shift from $d\approx2$ to $d\approx4$ dimensions is then discussed.Comment: 30 pages, 1 figure, minor typos in the text corrected, results unchanged. Version accepted for publication on Nuclear Physics

An Elliptic Generalization of Multiple Polylogarithms

We introduce a class of functions which constitutes an obvious elliptic generalization of multiple polylogarithms. A subset of these functions appears naturally in the \epsilon-expansion of the imaginary part of the two-loop massive sunrise graph. Building upon the well known properties of multiple polylogarithms, we associate a concept of weight to these functions and show that this weight can be lowered by the action of a suitable differential operator. We then show how properties and relations among these functions can be studied bottom-up starting from lower weights.Comment: 27 pages plus three appendices, v2: references added, typos corrected, accepted for publication on NP

Recursion relations for Hylleraas three-electron integral

Recursion relations for Hylleraas three-electron integral are obtained in a closed form by using integration by parts identities. Numerically fast and well stable algorithm for the calculation of the integral with high powers of inter-electronic coordinates is presented.Comment: 12 pages, requires RevTeX4, submitted to Phys. Rev.

BOKASUN: a fast and precise numerical program to calculate the Master Integrals of the two-loop sunrise diagrams

We present the program BOKASUN for fast and precise evaluation of the Master Integrals of the two-loop self-mass sunrise diagram for arbitrary values of the internal masses and the external four-momentum. We use a combination of two methods: a Bernoulli accelerated series expansion and a Runge-Kutta numerical solution of a system of linear differential equations

Iterated integrations of complete elliptic integrals

We study an elliptic generalization of multiple polylogarithms that appears naturally in the computation of the imaginary part of the two-loop massive sunrise graph with equal masses. The newly introduced functions fulfill non-homogeneous second order differential equations. As an important result, we introduce a concept of weight associated to the action of the second order differential operator and show how to classify the relations between the functions bottom up in their weight