Blade: A package for block-triangular form improved Feynman integrals decomposition

In this article, we present the package 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 Blade is remarkably competitive among existing reduction tools. Furthermore, the Blade package offers several distinctive features, including support for complex kinematic variables or masses, user-defined Feynman prescriptions for each propagator, and general integrands.

本文中，我们推出了Blade软件包，这是块三角形式改进费恩曼积分（Feynman integral）约化方法的首个实现版本。相较于普通的分部积分系统，块三角形式的方程数量在量级上大幅减少，可实现严格的分块求解，进而提升计算速度并降低资源消耗。我们阐明了用于推导块三角形式的相关算法及其具体实现。此外，我们还提出了用于求解费恩曼积分标准形式与对称关系、以及执行跨扇区约化的全新算法。我们针对各类前沿问题开展的基准测试表明，Blade在现有约化工具中具备显著的竞争力。此外，Blade软件包还具备多项独特特性：支持复运动学变量或质量、可为每个传播子自定义费恩曼规则，以及兼容通用被积函数。