The O(N) S-matrix Monolith

TL;DR

The paper studies the space of two-to-two S-matrices for massive 2D QFTs with O(N) symmetry and no bound states, constrained by unitarity, crossing, and analyticity. It reveals a rich convex structure dubbed the O(N) monolith, with a 2D slate arising at the crossing-symmetric point s*=2m^2; boundary points include integrable models like the free theory, the O(N) nonlinear sigma model, and periodic Yang-Baxter solutions. A dual convex minimization is developed, proving that boundary S-matrices generically saturate unitarity and explaining the zero duality gap; this duality also brackets the boundary and yields analytic handles on extremal S-matrices. The work highlights intriguing boundary phenomena (periodicity, fractal pole/zero structures) and lays groundwork for higher-dimensional generalizations and deeper connections to integrable structures.

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.

Figures (13)

• Figure 1: S-matrix monolith for $O(7)$ and $s_*=3m^2$. The best way to feel -- quite literally -- the various vertices, pre-vertices, edges and faces of the monolith is to 3D print it. We can easily detect nearly imperceptible vertices with one's fingertips fingertips, see also figure \ref{['lessSharp']} below. We attach an ancillary file 3dPrint.stl made out of a discretization of the monolith with more than 200,000 points (using the method of normals explained below) which can be directly printed or very efficiently visualized visualization. To generate such 3D printing file for the convex monolith is quite simple. We generate a huge list of points belonging to the monolith and then create the convex hull of all these points using Mathematica's built-in function ConvexHullMesh which can then be exported directly into an .stl file.
• Figure 2: Some features of the $O(N)$ monolith. Three arrows point to the integrable solutions corresponding to vertices (Free, NLSM) or pre-vertices (periodic YB) of the monolith. A fourth arrow points the yellow point corresponding to a constant solution which does not saturate unitarity. A line of simple (yet non-integrable) S-matrices connecting the two periodic Yang-Baxter solutions is highlighted in green. For each such special feature there is a mirror one simply related by $S_a\to -S_a$ which is a clear symmetry of the monolith.
• Figure 3: By construction, the minimum of the dual problem (\ref{['Fmin']}) puts a strict upper bound on the maximum of the original problem (\ref{['example']}). A priori the optimal minimum and the optimal maximum are separated by what is known as the duality gap as depicted in ( a). For convex problems the duality gap is zero and thus both problems describe the very same boundary of the S-matrix space, one converging from its interior, the other from its exterior, see ( b). With different ansatze we can thus rigorously bracket the optimal bound (in black in ( b)). Strictly speaking the previous statements should be qualified by the statement that both the dual and the primal problem ought to be feasible which is the case for us.
• Figure 4: Two-dimensional section of the monolith we call the $O(N)$ slate obtained at $s_*=2m^2$ ($\theta_*=i\pi/2$) in the $\sigma_i$ decomposition of \ref{['sigma_decomposition']} for $N=7$. In black we show the optimal bound to which the primal and dual problems converge respectively from below or above. Consistent S-matrices lie on the shaded region in grey. In blue (red) we present various bounds as we take periodic ansatze in the primal (dual) problem. From lighter to darker colors we have period $\tau=0,4,6,10.25$.
• Figure 5: Different maximization functionals to obtain the boundary of a certain region of the plane. In panel ( a) we fix one of the coordinates and maximize/minimize the other. In the second panel ( b) we fix a particular direction and perform a radial maximization, which useful for defining the faces of the convex space. Finally in ( c) we have the normal maximization where we have a uniform distribution of unit vectors and the maximization chooses the points where the normals are aligned with the unit vectors, resulting in a higher concentration of points in high curvature regions.