Principal Landau determinants

We reformulate the Landau analysis of Feynman integrals with the aim of advancing the state of the art in modern particle-physics computations. We contribute new algorithms for computing Landau singularities, using tools from polyhedral geometry and symbolic/numerical elimination. Inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ) on generalized Euler integrals, we define the principal Landau determinant of a Feynman diagram. We illustrate with a number of examples that this algebraic formalism allows to compute many components of the Landau singular locus. We adapt the GKZ framework by carefully specializing Euler integrals to Feynman integrals. For instance, ultraviolet and infrared singularities are detected as irreducible components of an incidence variety, which project dominantly to the kinematic space. We compute principal Landau determinants for the infinite families of one-loop and banana diagrams with different mass configurations, and for a range of cutting-edge Standard Model processes. Our algorithms build on the Julia package Landau.jl and are implemented in the new open-source package PLD.jl available at https://mathrepo.mis.mpg.de/PLD/.我们对费曼积分（Feynman integrals）的朗道分析进行了重新表述，以期推动现代粒子物理计算领域的前沿发展。我们借助多面体几何、符号与数值消元相关工具，提出了用于计算朗道奇点的全新算法。受格尔凡德、卡普兰诺夫与泽列维斯基（GKZ）关于广义欧拉积分的研究启发，我们定义了费曼图（Feynman diagram）的主朗道行列式。我们通过多个示例展示，该代数形式体系可用于计算朗道奇异轨迹的诸多组分。我们通过将欧拉积分严格特例化为费曼积分，对GKZ框架进行了适配调整。例如，紫外与红外奇点可被识别为关联簇（incidence variety）的不可约组分，这些簇在运动学空间上具有主导投影。我们针对具有不同质量构型的一圈费曼图与香蕉图（one-loop and banana diagrams）的无穷族，以及一系列前沿标准模型物理过程，计算了其主朗道行列式。我们的算法基于Julia软件包Landau.jl开发，并在全新开源软件包PLD.jl中实现，该软件包可从https://mathrepo.mis.mpg.de/PLD/ 获取。