A New Algorithm For The Generation Of Unitarity-Compatible Integration By Parts Relations

TL;DR

The paper tackles the challenge of generating complete sets of unitarity-compatible integration by parts (IBP) relations without doubled propagators in multi-loop Feynman integral reductions. It introduces a conceptually simple, linear-algebra-based algorithm inspired by LASyz to compute the necessary syzygies degree by degree, avoiding computationally intensive Gröbner bases. A detailed planar massless double-box example demonstrates that the method retrieves the known syzygy structure (one degree-1 and two degree-2) and yields unitarity-compatible IBP relations. The approach promises improved practicality and potential for high-performance, compiled implementations in multi-loop unitarity calculations.

Abstract

Many multi-loop calculations make use of integration by parts relations to reduce the large number of complicated Feynman integrals that arise in such calculations to a simpler basis of master integrals. Recently, Gluza, Kajda, and Kosower argued that the reduction to master integrals is complicated by the presence of integrals with doubled propagator denominators in the integration by parts relations and they introduced a novel reduction procedure which eliminates all such integrals from the start. Their approach has the advantage that it automatically produces integral bases which mesh well with generalized unitarity. The heart of their procedure is an algorithm which utilizes the weighty machinery of computational commutative algebra to produce complete sets of unitarity-compatible integration by parts relations. In this paper, we propose a conceptually simpler algorithm for the generation of complete sets of unitarity-compatible integration by parts relations based on recent results in the mathematical literature. A striking feature of our algorithm is that it can be described entirely in terms of straightforward linear algebra.