Principal Landau Determinants

TL;DR

This paper introduces a novel algebraic approach using polyhedral geometry and symbolic/numerical methods to compute Landau singularities in Feynman integrals, enhancing particle physics calculations.

Contribution

It develops new algorithms and the concept of principal Landau determinants, extending GKZ methods to analyze singularities in Feynman diagrams.

Findings

Successfully computed Landau determinants for various diagrams.

Detected ultraviolet and infrared singularities via algebraic components.

Implemented algorithms in open-source Julia packages.

Abstract

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/.