The O(N) S-matrix Monolith

TL;DR

This paper explores the space of two-to-two scattering matrices in 2D quantum field theories with O(N) symmetry, revealing a rich structure and identifying special integrable models at the boundaries.

Contribution

It introduces a convex optimization approach to analyze the space of S-matrices, uncovering geometric features and integrable models at the boundaries.

Findings

Identified vertices corresponding to integrable models.

Developed a convex dual minimization framework for S-matrix analysis.

Proved that generic S-matrices saturate unitarity constraints.

Abstract

We consider the scattering matrices of massive quantum field theories with no bound states and a global $O(N)$ symmetry in two spacetime dimensions. In particular we explore the space of two-to-two S-matrices of particles of mass $m$ transforming in the vector representation as restricted by the general conditions of unitarity, crossing, analyticity and $O(N)$ symmetry. We found a rich structure in that space by using convex maximization and in particular its convex dual minimization problem. At the boundary of the allowed space special geometric points such as vertices were found to correspond to integrable models. The dual convex minimization problem provides a novel and useful approach to the problem allowing, for example, to prove that generically the S-matrices so obtained saturate unitarity and, in some cases, that they are at vertices of the allowed space.