Landau Discriminants

Abstract

Scattering amplitudes in quantum field theories have intricate analytic properties as functions of the energies and momenta of the scattered particles. In perturbation theory, their singularities are governed by a set of nonlinear polynomial equations, known as Landau equations, for each individual Feynman diagram. The singularity locus of the associated Feynman integral is made precise with the notion of the Landau discriminant, which characterizes when the Landau equations admit a solution. In order to compute this discriminant, we present approaches from classical elimination theory, as well as a numerical algorithm based on homotopy continuation. These methods allow us to compute Landau discriminants of various Feynman diagrams up to 3 loops, which were previously out of reach. For instance, the Landau discriminant of the envelope diagram is a reducible surface of degree 45 in the three-dimensional space of kinematic invariants. We investigate geometric properties of the Landau discriminant, such as irreducibility, dimension and degree. In particular, we find simple examples in which the Landau discriminant has codimension greater than one. Furthermore, we describe a numerical procedure for determining which parts of the Landau discriminant lie in the physical regions. In order to study degenerate limits of Landau equations and bounds on the degree of the Landau discriminant, we introduce Landau polytopes and study their facet structure. Finally, we provide an efficient numerical algorithm for the computation of the number of master integrals based on the connection to algebraic statistics. The algorithms used in this work are implemented in the open-source Julia package Landau.jl available at https://mathrepo.mis.mpg.de/Landau/.

67 pages, 8 figures, journal version

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• Loop-by-loop Differential Equations for Dual (Elliptic) Feynman Integrals
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• Exposing the threshold structure of loop integrals
• Towards Analytic Structure of Feynman Parameter Integrals with Rational Curves
• Natural Boundaries for Scattering Amplitudes
• Feynman Integrals of Grassmannians
• Hierarchies in relative Picard-Lefschetz theory
• Probing multi-particle unitarity with the Landau equations
• Cutkosky's Theorem for Massive One-Loop Feynman Integrals -- Part 1