Speaker
Description
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 singularity structure is governed by nonlinear polynomial systems known as Landau equations. In this work we introduce several tools from computational algebraic geometry to solving Landau equations for any individual Feynman diagram. Singularity locus of the associated Feynman integral is made precise with the notion of the Landau discriminant. In order to solve it, we discuss two basic approaches from classical elimination theory, as well as a new algorithm based on tools from numerical nonlinear algebra such as homotopy continuation. These methods allow us to compute Landau discriminants of various Feynman diagrams up to $3$ loops, including the envelope diagram whose Landau discriminant is a reducible surface of degree $45$ in the space of kinematic invariants $\mathbb{P}^3$. In addition, we construct Landau polytopes in order to study degenerations of Landau equations and exemplify them on the family of banana diagrams. As a byproduct, we provide an efficient algorithm for the computation of the number of master integrals based on the connection to algebraic statistics.
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