The R-matrix bootstrap for the 2d O(N) bosonic model with a boundary

TL;DR

This work extends the S-matrix bootstrap to 1+1-dimensional theories with an $O(N)$ boundary by focusing on reflection matrices (R-matrices) and mapping the space of allowed R-matrices for a fixed bulk S-matrix. It reveals that integrable R-matrices often appear at boundary vertices, while extended analyticity can contract the allowed region and reveal new integrable points, including a new analytic R-matrix for the periodic Yang-Baxter solution. The authors implement a numerical bootstrap on strips with periodicity, reproduce known integrable R-matrices for the $O(N)$ non-linear sigma model and pYB, and derive analytic forms using the Yang-Baxter equation and $H_\nu$-functions. They also formulate a dual problem, showing how unitarity saturation arises and how extended analyticity modifies the dual, providing a robust framework for exploring boundary S-matrix data with potential extensions to bound-state sectors and supersymmetric models.

Abstract

The S-matrix bootstrap is extended to a 1+1d theory with $O(N)$ symmetry and a boundary in what we call the R-matrix bootstrap since the quantity of interest is the reflection matrix (R-matrix). Given a bulk S-matrix, the space of allowed R-matrices is an infinite dimensional convex space from which we plot a two dimensional section given by a convex domain on a 2d plane. In certain cases, at the boundary of the domain, we find vertices corresponding to integrable R-matrices with no free parameters. In other cases, when there is a one-parameter family of integrable R-matrices, the whole boundary represents integrable theories. We also consider R-matrices which are analytic in an extended region beyond the physical cuts, thus forbidding poles (resonances) in that region. In certain models, this drastically reduces the allowed space of R-matrices leading to new vertices that again correspond to integrable theories. We also work out the dual problem, in particular in the case of extended analyticity, the dual function has cuts on the physical line whenever unitarity is saturated. For the periodic Yang-Baxter solution that has zero transmission, we computed the R-matrix initially using the bootstrap and then derived its previously unknown analytic form.

Figures (12)

• Figure 1: Pictorial description of the $1\rightarrow 1$ R-matrix describing the amplitude for a particle to bounce from the wall possibly changing its identity $a\rightarrow b$. A double Wick rotation relates this process to pair production from an initial boundary state $\hbox{$| B \rangle$}$. In the second case we can define a bulk S-matrix using the usual asymptotic states.
• Figure 2: Physical region of the reflection process ($\mathrm{Re}\, \varepsilon \ge 0$). The positive real axis corresponds to the physical reflection ($\varepsilon \in \mathbb{R}_{\ge m}$) which is the boundary value of an analytic function in the physical region (shaded). Crossing can be imposed on the upper line, this is the minimal region where crossing gives a constraint. Boundary bound states could appear as poles on the imaginary axis (dashed line) but we do not allow them here.
• Figure 3: Extended region of analyticity. We considered two cases: $b_1=0$, $\frac{\pi}{2}\le b_2\le \pi$ and $b_1=\pi-b_2$, $b_2>\pi$. It might seem that in the first case analyticity is guaranteed by crossing but this is not so because of the non-trivial crossing equation (\ref{['cross_unit']}).
• Figure 4: Plot of the $(R_1(\theta_1),R_2(\theta_1))$, $\theta_1=0.2i$, allowed region for the NLSM with $N=6$, $k=1$ and various analytic regions $b_1=0$, $b_2=\frac{\pi}{2}, 0.9\pi$, and $b_1=-0.1\pi$, $b_2=1.1\pi$. The vertex at the center of the red circle (see rotated and rescaled inset) is the R-matrix in eq.(\ref{['a14']}) with $N=6$, $k=1$. A new vertex appears in the smaller regions, namely the one at the center of the purple circle that corresponds to a solution with $R_1=R_2$, or $N=6$, $k=0$.
• Figure 5: Plot of $R_1(\theta)$ and $R_2(\theta)$ on the real axis. Figures (a) and (b) correspond to the red dashed vertex of fig.\ref{['NLSM_allowed_regions']}. It agrees precisely with Dirichlet R-matrix. Figure (c) is the purple dashed vertex of fig.\ref{['NLSM_allowed_regions']} and agrees with the Neumann R-matrix.