New approach to evaluation of multiloop Feynman integrals: The Gegenbauer polynomial x-space technique

Abstract

We formulate the Gegenbauer polynomial x-space technique for evaluating dimensionally regularized massless Feynman diagrams. We show that the analytical evaluation of counterterms in the minimal subtraction scheme at the 3-loop level in any model and for any composite operator is really possible. We calculate exactly in 4 − 2ε dimensions the 2-loop massless propagator-type integrals needed in computing 3-loop counterterms. At the 4-loop level the larger part of diagrams, consisting of those with a propagator-type subgraph and some others, has proved no more difficult. We also explain how calculations can be strikingly simplified in a certain subtraction scheme - the G-scheme. As an application of the developed methods we calculate the 3-loop contribution to σ tot(e +e − →hadrons) and briefly discuss the results together with their implications for the QED β-function.