Vol. 12, No. 2, 2021

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Likelihood equations and scattering amplitudes

Bernd Sturmfels and Simon Telen

Vol. 12 (2021), No. 2, 167–186
DOI: 10.2140/astat.2021.12.167
Abstract

We relate scattering amplitudes in particle physics to maximum likelihood estimation for discrete models in algebraic statistics. The scattering potential plays the role of the log-likelihood function, and its critical points are solutions to rational function equations. We study the ML degree of low-rank tensor models in statistics, and we revisit physical theories proposed by Arkani-Hamed, Cachazo and their collaborators. Recent advances in numerical algebraic geometry are employed to compute and certify critical points. We also discuss positive models and how to compute their amplitudes.

Keywords
likelihood equations, scattering amplitudes, numerical nonlinear algebra
Mathematical Subject Classification
Primary: 62R01, 65H14
Milestones
Received: 9 December 2020
Revised: 29 April 2021
Accepted: 18 May 2021
Published: 13 December 2021
Authors
Bernd Sturmfels
Max Planck Institute for Mathematics in the Sciences
Leipzig
Germany
Department of Mathematics
University of California, Berkeley
Berkeley, CA
United States
Simon Telen
Max Planck Institute for Mathematics in the Sciences
Leipzig
Germany