Reduction of Feynman graph amplitudes to a minimal set of basic integrals

TL;DR

The paper tackles the problem of reducing massive multi-loop Feynman integrals to a minimal set of basic integrals by mapping tensor integrals to scalar integrals in shifted dimensions, deriving generalized recurrence relations, and recasting them as linear PDEs. It then applies G. Reid's Standard Form algorithm to identify a minimal, finite basis and to determine the solution-space dimension, linking leading derivatives to the recurrence structure and parametric integrals to the basis. This framework provides a principled, potentially scalable path for analytical and semi-analytical evaluation of complex Feynman diagrams, with particular relevance to gauge theories and precision SM calculations. Overall, it offers a systematic, PDE-based methodology to achieve tensor reduction, basis selection, and, ultimately, reliable radiative corrections in high-energy physics.

Abstract

An algorithm for the reduction of massive Feynman integrals with any number of loops and external momenta to a minimal set of basic integrals is proposed. The method is based on the new algorithm for evaluating tensor integrals, representation of generalized recurrence relations for a given kind of integrals as a linear system of PDEs and the reduction of this system to a standard form using algorithms proposed by G. Reid. Basic integrals reveal as parametric derivatives of the system in the standard form and the number of basic integrals in the minimal set is determined by the dimension of the solution space of the system of PDEs.