A note on the S-matrix bootstrap for the 2d O(N) bosonic model

TL;DR

This paper demonstrates that the 2d O(N) bosonic integrable model can be identified as a vertex of a convex space of S-matrices by maximizing linear functionals, offering a new numerical approach independent of integrability constraints.

Contribution

It introduces a novel method to locate integrable models as vertices in the convex space of S-matrices through linear functional maximization, bypassing traditional integrability equations.

Findings

The model resides at a vertex of the convex space defined by unitarity and crossing.

Maximizing linear functionals can reproduce the analytical solution without integrability.

The approach is potentially applicable to other theories without continuous parameters.

Abstract

In this work we apply the S-matrix bootstrap maximization program to the 2d bosonic O(N) integrable model which has N species of scalar particles of mass m and no bound states. Since in previous studies theories were defined by maximizing the coupling between particles and their bound states, the main problem appears to be to find what other functional can be used to define this model. Instead, we argue that the defining property of this integrable model is that it resides at a vertex of the convex space determined by the unitarity and crossing constraints. Thus, the integrable model can be found by maximizing any linear functional whose gradient points in the general direction of the vertex, namely within a cone determined by the normals to the faces intersecting at the vertex. This is a standard problem in applied mathematics, related to semi-definite programming and solvable by fast available numerical algorithms. The information provided by the numerical solution is enough to reproduce the known analytical solution without using integrability, namely the Yang-Baxter equation. This situation seems quite generic so we expect that other theories without continuous parameters can also be found by maximizing linear functionals in the convex space of allowed S-matrices.