AN ALGORITHM FOR SMALL MOMENTUM EXPANSION OF FEYNMAN DIAGRAMS
O. V. TARASOV
TL;DR
The paper introduces a recurrence-relations algorithm for the small-momentum expansion of Feynman diagrams, enabling efficient computation of Taylor coefficients for propagator and two-loop vertex diagrams without relying on explicit differentiation. By factoring external momenta and reducing to master integrals, the method achieves large computational savings and is validated through two-loop applications, enhanced by conformal mapping and Padé approximants to extract precise results near and across branch cuts. The approach yields accurate coefficients with significantly reduced CPU time, highlighting its potential for high-precision multi-loop evaluations and applications to anomalous dimensions and structure-function moments. Overall, it offers a scalable, implementable framework that improves both speed and precision in small-momentum expansions of complex diagrams.
Abstract
An algorithm for obtaining the Taylor coefficients of an expansion of Feynman diagrams is proposed. It is based on recurrence relations which can be applied to the propagator as well as to the vertex diagrams. As an application, several coefficients of the Taylor series expansion for the two-loop propagator and two-loop non-planar vertex diagrams are calculated. The results of the numerical evaluation of these diagrams using conformal mapping and Pade approximants are given.