Blade: A package for block-triangular form improved Feynman integrals decomposition
Xin Guan, Xiao Liu, Yan-Qing Ma, Wen-Hao Wu
TL;DR
Blade introduces a public implementation of block-triangular form improved Feynman integral decomposition to dramatically reduce the size of IBP systems and enable block-by-block reduction. It combines a rigorous framework of IBP/LIs/zero sectors, finite-field techniques, and syzygy-based optimizations with a sophisticated block-triangular search and adaptive strategies to automate master-integral reduction. The method demonstrates substantial performance gains across diverse multi-loop benchmarks, enabling reductions that were previously computationally prohibitive. The work also outlines extensive tooling for canonical forms and discretized symmetries, with practical features for complex kinematics and general integrands, and sketches several avenues for future enhancements and broader integration with other symbolic-numerical packages.
Abstract
In this article, we present the package {\tt Blade} as the first implementation of the block-triangular form improved Feynman integral reduction method. The block-triangular form has orders of magnitude fewer equations compared to the plain integration-by-parts system, allowing for strictly block-by-block solutions. This results in faster evaluations and reduced resource consumption. We elucidate the algorithms involved in obtaining the block-triangular form along with their implementations. Additionally, we introduce novel algorithms for finding the canonical form and symmetry relations of Feynman integrals, as well as for performing spanning-sector reduction. Our benchmarks for various state-of-the-art problems demonstrate that {\tt Blade} is remarkably competitive among existing reduction tools. Furthermore, the {\tt Blade} package offers several distinctive features, including support for complex kinematic variables or masses, user-defined Feynman prescriptions for each propagator, and general integrands.