Feynman Integral Reduction using Syzygy-Constrained Symbolic Reduction Rules

TL;DR

This paper introduces a novel algorithm for efficiently reducing complex multi-loop Feynman integrals using syzygy equations, symbolic manipulation, and targeted linear systems, significantly speeding up amplitude calculations.

Contribution

The paper presents a new syzygy-based algorithm for IBP reduction that handles high-power integrals with symbolic dependence, improving computational efficiency in multi-loop amplitude evaluations.

Findings

Successfully applied to double box and pentabox integrals with high complexity.

Achieved faster IBP reduction in black hole binary scattering amplitude calculations.

Demonstrated effectiveness on non-trivial two-loop Feynman integrals.

Abstract

We present a new algorithm for integration-by-parts (IBP) reduction of Feynman integrals with high powers of numerators or propagators, a demanding computational step in evaluating multi-loop scattering amplitudes. The algorithm starts with solving syzygy equations in individual sectors to produce IBP operators that turn seed integrals into IBP equations without artificially raised propagator powers. The IBP operators are expressed in terms of index-shift operators and number operators. We perform row reduction to systematically reshuffle the IBP operators and expose reduction rules with symbolic dependence on the powers of propagators and numerators. When this is insufficient, we produce more symbolic reduction rules by directly solving the linear system of IBP equations in which some propagator/numerator powers are kept symbolic. This linear system is kept small, as the equations are generated from a small set of seed integrals in the neighborhood of the target integral. We stress-test our algorithm against two highly non-trivial examples, namely rank-20 integrals for the double box with an external mass and the massless pentabox. As an application, we revisit the IBP reduction in a calculation of scattering amplitudes for spinning black hole binary systems, which involves two-loop Feynman integrals with complexity greater than 20, and achieve much faster IBP reduction than that of the original calculation.