ar X iv : h ep - p h / 99 05 37 9 v 1 1 8 M ay 1 99 9 Recursion relations for two - loop self

A set of recurrence relations for on-shell two-loop self-energy diagrams with one mass is presented, which allows to reduce the diagrams with arbitrary indices (powers of scalar propagators) to a set of the master integrals. The SHELL2 package is used for the calculation of special types of diagrams. A method of calculation of higher order ε-expansion of master integrals is demonstrated. Nowadays e + e −-experiments are sensitive to multiloop radiative corrections. The transverse part of renormalized gauge boson self-energies on mass shell enter a wide class of low energy observables like ∆ρ, ∆r, etc. Keeping in mind this physical application we elaborate a FORM [1] based package 1 [3] for analytical calculations of on-shell two-loop self-energy diagrams with one mass 2. All possible diagrams of given type are shown on Fig.1. The diagrams implemented in the package SHELL2 [5] (ON3, ON2 in our notations) and those considered in detail in Ref.[6] (F00000, V0000, J001, J000) are not discussed here. In contrary to Ref.[7] we used only recurrence relations obtained from the integration by part method [6] without shifting the dimension of space-time. Here we apply the triangle rule for arbitrary masses and external momentum given in Ref.[8]. We are working in Euclidean space-time with dimension N = 4 − 2ε. The general prototype involves arbitrary integer powers of the scalar denominators c L = k 2