
TL;DR
This paper introduces new computational methods and algorithms for analyzing Landau discriminants in quantum field theory, enabling the study of complex Feynman diagrams and their singularities up to three loops.
Contribution
It presents classical elimination theory and homotopy continuation algorithms to compute Landau discriminants, along with geometric and numerical analyses of their properties.
Findings
Computed Landau discriminants for diagrams up to 3 loops.
Identified geometric properties such as reducibility and codimension.
Developed an open-source Julia package for these computations.
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…
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