New Method of Massive Feynman Diagrams Calculation

Abstract

A new method of massive Feynman integrals calculation which is based on the rule of integration by parts is presented. This rule is expanded to the massive case. By applying this rule, we obtain a differential equation with respect to the mass for the initial diagram. The right-hand side of the equation contains simpler diagrams (i.e., containing only loops, not chains). This can be done by applying the procedure consecutively. These loops can be calculated either by the standard Feynman-parameter procedure or by a procedure which decreases the number of loops step-by-step. We demonstrate the capacities of this method for various complicated diagrams and make an attempt to analyze other possible massive Feynman diagrams calculations.

QED vertex form factors at two loops

“…(12)(13)(14), expanded up to the first order in (D − 4). The Feynman diagrams involved are those in Fig.…”

Master integrals for the 2-loop QCD virtual corrections to the forward–backward asymmetry

“…17 of them were already calculated in [7]. We present here the analytical evaluation of the remaining 18, obtained by means of the differential equations method [13,14,15] or, when all the propagators are massless, via direct integration with the Feynman parameters. The Master Integrals are Laurent-expanded around D = 4 and the coefficients of the Laurent-expansion are then expressed in terms of 1-dimensional harmonic polylogarithms (HPLs) [16,17].…”

Calculation of Feynman diagrams from their small momentum expansion

“…In particular two-loop calculations with nonzero masses became relevant [2]. While in the one-loop approach there exists a systematic way of performing these calculations [3], in the two-loop case there does not exist such a developed technology and only a series of partial results were obtained [4], [5] but no systematic approach w as formulated. In the present w ork we demonstrate exactly such an approach.…”