Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space
Dominik Bendle, Janko Boehm, Wolfram Decker, Alessandro Georgoudis, Franz-Josef Pfreundt, Mirko Rahn, Pascal Wasser, Yang Zhang
TL;DR
This work introduces an algebro-geometric IBP-reduction framework that fuses the Baikov representation with module-intersection techniques to produce compact, double-propagator-free IBP systems and solves them via sparse linear algebra and rational interpolation. By modeling computations as Petri nets in the GPI-Space workflow manager and outsourcing the heavy lifting to Singular, the authors achieve automated, massively parallel reductions, demonstrated on the challenging two-loop five-point nonplanar double pentagon. A key result is the significant reduction in IBP-coefficient sizes when converting to a dlog (uniform transcendental) basis, alongside extensive cross-checks against existing tools. The approach promises scalable NNLO computations and provides a flexible pathway for applying algebraic-geometry techniques to large-scale amplitude problems.
Abstract
We introduce an algebro-geometrically motived integration-by-parts (IBP) reduction method for multi-loop and multi-scale Feynman integrals, using a framework for massively parallel computations in computer algebra. This framework combines the computer algebra system Singular with the workflow management system GPI-Space, which is being developed at the Fraunhofer Institute for Industrial Mathematics (ITWM). In our approach, the IBP relations are first trimmed by modern algebraic geometry tools and then solved by sparse linear algebra and our new interpolation methods. These steps are efficiently automatized and automatically parallelized by modeling the algorithm in GPI-Space using the language of Petri-nets. We demonstrate the potential of our method at the nontrivial example of reducing two-loop five-point nonplanar double-pentagon integrals. We also use GPI-Space to convert the basis of IBP reductions, and discuss the possible simplification of IBP coefficients in a uniformly transcendental basis.