Symbolic syzygy-constrained reduction rules for Feynman integrals and the LoopIn framework

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 allows us to avoid a large intermediate system of equations and instead focus on applying direct reduction rules to the integrals. We demonstrate the application of our algorithm with some highly non-trivial examples, namely rank-20 integrals for the double box with an external mass and the massless pentabox. We also achieve much faster IBP reduction for an example of scattering amplitudes for spinning black hole binary systems. Finally, we present LoopIn, a modular framework for automating multi-loop calculations, where the IBP techniques described here can be interfaced.

Figures (6)

• Figure 1: Tower of Sectors one must consider when the top sectors are $(2,1,1,0,0)$ and $(1,2,1,0,0)$. This drawing is schematic and the horizontal levels here do not correlate with the weighting described later. The relevant information is contained within the arrows, denoting the subsector inheritance hierarchy.
• Figure 2: A flow chart describing the algorithm for generating reduction rules
• Figure 3: Two Loop Box
• Figure 4: Massless Pentabox
• Figure 5: Post-Minkowskian Non-Planar Double Box