Likelihood Equations and Scattering Amplitudes

TL;DR

This paper connects scattering amplitudes in physics to maximum likelihood estimation in algebraic statistics, exploring mathematical properties and computational methods for analyzing these models.

Contribution

It introduces a novel framework relating scattering amplitudes to likelihood equations, studying ML degrees of tensor models and applying algebraic geometry techniques.

Findings

Relation between scattering amplitudes and likelihood functions.

Computation of critical points using numerical algebraic geometry.

Analysis of positive models and string amplitude computations.

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 string amplitudes.