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

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.

20 pages; My apologies to Krzysztof Kajda for misspelling his name in v1; in v3: the labeling of the variables in (4.5) and eqs. (4.20) and (4.21) was adjusted to match the notation used in the rest of Section 4. I thank York Schroeder for pointing out the notational inconsistency

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• Algorithm FIRE -- Feynman Integral REduction
• On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes
• Maximal Unitarity at Two Loops
• An Overview of Maximal Unitarity at Two Loops

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