Module Intersection for the Integration-by-Parts Reduction of Multi-Loop Feynman Integrals

TL;DR

This work addresses the bottleneck of integrating-by-parts (IBP) reductions for multi-loop Feynman integrals by employing a module-intersection approach within the Baikov representation to produce trimmed IBP systems that can restrict propagator degrees and avoid double propagators. The authors combine algebraic-geometric module intersections with interpolation-based Gaussian elimination, implemented in the Singular-GPI-Space framework, and leverage a plugin-enabled Petri-net workflow for scalable parallel computation. A non-trivial two-loop, five-point non-planar example demonstrates the method: starting from 113 irreducible integrals (108 masters after symmetries), the approach uses 11 spanning cuts to generate IBPs without double propagators, and then achieves a dramatic reduction in coefficient sizes by switching to a $d\log$ basis, with further compression via Leinartas' algorithm. The results show promise for handling higher-point/higher-loop reductions and hint at broad applicability to related problems in mathematical physics, such as integrable models and Grassmannian structures.

Abstract

In this manuscript, which is to appear in the proceedings of the conference "MathemAmplitude 2019" in Padova, Italy, we provide an overview of the module intersection method for the the integration-by-parts (IBP) reduction of multi-loop Feynman integrals. The module intersection method, based on computational algebraic geometry, is a highly efficient way of getting IBP relations without double propagator or with a bound on the highest propagator degree. In this manner, trimmed IBP systems which are much shorter than the traditional ones can be obtained. We apply the modern, Petri net based, workflow management system GPI-Space in combination with the computer algebra system Singular to solve the trimmed IBP system via interpolation and efficient parallelization. We show, in particular, how to use the new plugin feature of GPI-Space to manage a global state of the computation and to efficiently handle mutable data. Moreover, a Mathematica interface to generate IBPs with restricted propagator degree, which is based on module intersection, is presented in this review.

Figures (5)

• Figure 1: Transition firing until it is disabled.
• Figure 2: Minimal Petri nets illustrating parallelisms
• Figure 3: Conflict of two transitions leading to different program flows.
• Figure 4: Updated and simplified Petri net incorporating plugin system.
• Figure 5: Two-loop five-point nonplanar double pentagon diagram with inversed propagators $z_i$.