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

TL;DR

The paper demonstrates that the 2d O(N) bosonic S-matrix, which has no bound states, can be reconstructed by treating the space of allowed S-matrices as a convex set bounded by unitarity and crossing and by maximizing linear functionals to locate a vertex. This vertex corresponds to the integrable model, reproduceable without invoking the Yang–Baxter equation, as the numerics align with the known analytic solution and with CDD-dressed variants. The results reveal an infinite family of functionals that yield the same integrable S-matrix and illuminate a broader principle: theories without continuous parameters may appear as boundary vertices in S-matrix space, suggesting a general method for discovering such theories via convex optimization. The work also clarifies how saturating unitarity via CDD factors and exploring other vertices enriches the boundary structure, with potential implications for applying similar ideas to higher-dimensional theories, including QCD.

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.

Figures (9)

• Figure 1: Different variables used to write the $S$-matrices. The first one $s$ is the usual Mandelstam variable and the other two are defined in eq.(\ref{['stheta']}).
• Figure 2: Schematic representation of the convex space of allowed S-matrices. The non-linear sigma model is at a vertex like $A$ where there are no directions such that at the linearized level satisfy the constraints in both directions. At a point like $B$ there are tangents that are parallel to the boundary and appear as zero modes of the linearized constraints. The dotted lines represent the linearized constraints described by $V^C_{B} \xi_B \le 0$.
• Figure 3: The comparison of the maximization results with the exact integrable model for $N=4$. We use $200$ points for interpolation and choose an initial functional $F=\mathrm{Re}[-S_1(\theta_0)-\alpha S_2(\theta_0)]$ with $\theta_0\simeq i(z_0=0.3i)$ and $\alpha=100$. Here we plot the real and imaginary parts of $S_I$, $S_+$ and $S_-$ on the physical line where the dots indicate maximization results while the curves are the exact integrable model.
• Figure 4: The comparison of the maximization results with the exact integrable model for $N=6$. We use $200$ points for interpolation and choose an initial functional $F=\mathrm{Re}[S_1(\theta_0)-\alpha S_2(\theta_0)]$ with $\theta_0\simeq i(z_0=0.3i)$ and $\alpha=7.5$. Here we plot the real and imaginary parts of $S_I$, $S_+$ and $S_-$ on the physical line where the dots indicate maximization results while the curves are the exact integrable model.
• Figure 5: The comparison of the maximization results with the exact integrable model for $N=20$. We use $200$ points for interpolation and choose an initial functional $F=\mathrm{Re}[S_1(\theta_0)-\alpha S_2(\theta_0)]$ with $\theta_0\simeq i(z_0=0.3i)$ and $\alpha=3$. Here we plot the real and imaginary parts of $S_I$, $S_+$ and $S_-$ on the physical line where the dots indicate maximization results while the curves are the exact integrable model.