Computation of Gröbner Bases for Two-Loop Propagator Type Integrals

TL;DR

This work applies Gröbner basis methods to two-loop propagator-type Feynman integrals with arbitrary masses and external momentum by transforming IBP recurrence relations into PDEs and dimension-shift relations. It computes Gröbner bases for the sunset ($J_3^{d}$), $V^{d}$, and $F^{d}$ topologies using Maple/Rif, obtaining finite bases that reduce higher-power integrals to a small set of master integrals and tadpole products, while exposing the structure of dimension-shift relations (e.g., $J_3^{d+2}$, $V^{d+2}$, $F^{d+2}$). The results demonstrate that existing software can handle these bases, enabling practical reduction workflows, though challenges remain for kinematic configurations where Gram determinants vanish; the authors discuss storage of bases for reuse and plan extensions to vertex integrals and robust handling of zero-denominator cases.

Abstract

The Gröbner basis technique for calculating Feynman diagrams proposed in [O.V. Tarasov, Acta Physica Polonica, v. B29 (1998) 2655] is applied to the two-loop propagator type integrals with arbitrary masses and momentum. We describe the derivation of Gröbner bases for all integrals with 1PI topologies and present elements of the Gröbner bases.