On the reduction of Feynman integrals to master integrals

TL;DR

This paper addresses automatic reduction of Feynman integrals to master integrals by introducing the s-bases method, a sector-aware adaptation of Gröbner-based reduction. It recasts IBP relations as a left ideal in an operator algebra and builds sector-specific bases (s-bases) to reduce any integral to masters and lower sectors. It also outlines a hybrid strategy that combines s-bases with Laporta's algorithm (FIRE) to handle all sectors efficiently, including applications to multi-loop (up to three-loop) calculations. The approach promises improved automation and scalability for complex multi-loop reductions and aims at public release of the implementation.

Abstract

The reduction of Feynman integrals to master integrals is an algebraic problem that requires algorithmic approaches at the modern level of calculations. Straightforward applications of the classical Buchberger algorithm to construct Groebner bases seem to be inefficient. An essential modification designed especially for this problem has been worked out. It has been already applied in two- and three-loop calculations. We are also suggesting to combine our method with the Laporta's algorithm.