标题： 用D-模和热带几何评估费曼积分 显示英文标题

标题： Evaluating Feynman Integrals Using D-modules and Tropical Geometry

摘要： 费曼积分在现代散射振幅研究计划中起着核心作用。因此，改进我们计算费曼积分的方法将增强我们比较粒子加速器（如大型强子对撞机）的理论预测与实验数据的能力。受此激励，本手稿旨在研究与费曼积分相关的数学概念。具体而言，我们提出了数值和解析算法来计算费曼积分。内容分为三个部分。第一部分专注于微分方程组（DEQs）方法用于计算费曼积分。一个原本令人望而生畏的积分表达式被转化为解决一组微分方程组的相对简单任务。我们使用该技术来评估与暗物质探测相关的两圈费曼积分族。第二部分将费曼积分的微分方程研究置于D-模框架内，这是代数研究偏微分方程的自然语言。特别强调了一个名为GKZ系统的特定D-模，这是一组高阶偏微分方程，它们消除了费曼积分的一个广义版本。在将广义积分匹配到适当的费曼积分过程中，我们发现了一种用对数级数表示后者的算法。第三部分开发了一种数值积分算法。它结合了蒙特卡洛抽样与热带几何，后者是代数几何的一个特殊分支，研究“分段线性”多项式。费曼的i*epsilon规则通过路径变形融入算法中。我们提供了一个名为Feyntrop的开源程序，实现了这个算法，并用它在物理相空间的1-5圈和0-5腿上数值计算费曼积分。 显示英文摘要

摘要： Feynman integrals play a central role in the modern scattering amplitudes research program. Advancing our methods for evaluating Feynman integrals will, therefore, strengthen our ability to compare theoretical predictions with data from particle accelerators such as the Large Hadron Collider. Motivated by this, the present manuscript purports to study mathematical concepts related to Feynman integrals. In particular, we present both numerical and analytical algorithms for the evaluation of Feynman integrals. The content is divided into three parts. Part I focuses on the method of DEQs for evaluating Feynman integrals. An otherwise daunting integral expression is thereby traded for the comparatively simpler task of solving a system of DEQs. We use this technique to evaluate a family of two-loop Feynman integrals of relevance for dark matter detection. Part II situates the study of DEQs for Feynman integrals within the framework of D-modules, a natural language for studying PDEs algebraically. Special emphasis is put on a particular D-module called the GKZ system, a set of higher-order PDEs that annihilate a generalized version of a Feynman integral. In the course of matching the generalized integral to a Feynman integral proper, we discover an algorithm for evaluating the latter in terms of logarithmic series. Part III develops a numerical integration algorithm. It combines Monte Carlo sampling with tropical geometry, a particular offspring of algebraic geometry that studies "piecewise-linear" polynomials. Feynman's i*epsilon-prescription is incorporated into the algorithm via contour deformation. We present an open-source program named Feyntrop that implements this algorithm, and use it to numerically evaluate Feynman integrals between 1-5 loops and 0-5 legs in physical regions of phase space.