Groebner Bases in Perturbative Calculations

TL;DR

The paper tackles the challenge of reducing multiloop loop integrals to master integrals when propagator powers are symbolic. It reframes integration-by-parts recurrences as linear finite-difference equations in indices and applies Gröbner-bases theory to obtain a universal reduction framework, enabling systematic discovery of master integrals and explicit reductions to them. A one-loop example demonstrates the method, yielding a small master set and illustrating how GB leading terms determine which shifted integrals vanish. The work also discusses computational challenges and proposes leveraging involutive GB algorithms and parallelism to manage the exponential complexity and parametric coefficients inherent in multiloop problems.

Abstract

In this paper we outline the most general and universal algorithmic approach to reduction of loop integrals to basic integrals. The approach is based on computation of Groebner bases for recurrence relations derived from the integration by parts method. In doing so we consider generic recurrence relations when propagators have arbitrary integer powers treated as symbolic variables (indices) for the relations.