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

Published for SISSA by Springer Received: January 28, 2021 Accepted: March 7, 2021 Published: April 12, 2021 Martin Kruczenskia,b and Harish Muralia a Department of Physics and Astronomy, Purdue Quantum Science and Engineering Institute (PQSEI), Purdue University, W. Lafayette, IN, U.S.A. b Purdue Quantum Science and Engineering Institute (PQSEI), Purdue University, W. Lafayette, IN, U.S.A. E-mail: , 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. Keywords: Boundary Quantum Field Theory, Field Theories in Lower Dimensions, Integrable Field Theories, Nonperturbative Effects ArXiv ePrint: 2012.15576 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP04(2021)097 JHEP04(2021)097 The R-matrix bootstrap for the 2d O(N ) bosonic model with a boundary Contents 1 1 3 4 5 7 7 2 Numerics 2.1 NLSM, diagonal R-matrix 2.2 NLSM, block diagonal R-matrix 2.3 pYB, diagonal R-matrix 9 10 10 10 3 The dual problem 3.1 Dual problem in the physical region 3.2 Dual to the extended analyticity problem 13 13 18 4 Conclusions 19 A A useful function 19 B Discussion of numerical approach 21 C Free bulk theory 22 1 1.1 Introduction The S-matrix and R-matrix bootstrap programs New insights were recently found on the old idea [1, 2] of determining the S-matrix directly from its analytic structure, symmetries, crossing, and unitarity. This certainly works in two dimensional integrable theories but only after using the factorization constraint, namely the Yang-Baxter equation. Without that, those constraints are not enough to completely determine the S-matrix. However, recently it was found that maximizing the coupling between particles and their bound states led to well-known theories such as a subsector of the sine-Gordon model. It can be also applied to 3+1 dimensional theories, and multiple amplitudes [3–6]. The main physical argument is that, when the spectrum of bound states is fixed, there is a limit on the value of the coupling since stronger couplings will lead to more bound states. This is a very powerful idea, namely that certain theories lay at –1– JHEP04(2021)097 1 Introduction 1.1 The S-matrix and R-matrix bootstrap programs 1.2 The 2d O(N ) bosonic model, general properties and exact S-matrices 1.3 The 2d O(N ) bosonic model, reflection matrices 1.3.1 NLSM, diagonal R-matrix 1.3.2 NLSM, block diagonal R-matrix 1.3.3 pYB, diagonal R-matrix Motivated by this, we consider the two dimensional bosonic O(N ) model with a boundary [33–36] and compute the reflection matrix (R-matrix). The procedure is similar. We impose all constraints of analyticity, crossing, and unitarity and map the allowed space of R-matrices looking for special points at the boundary of the space. A difference is that crossing depends on the bulk S-matrix that we have to specify initially. It might be interesting to find the S-matrix and R-matrix simultaneously, but the constraints are non-linear. In fact, it seems more straightforward to compute first the S-matrix and then the R-matrix. In any case, this is the procedure we follow here and find the same variety of phenomena previously discussed for the S-matrix. When an integrable R-matrix exists with no free parameters it usually appears at a vertex of the allowed space. If this is not the case, we apply a new procedure that we call extended analyticity where we extend the analytic properties of the functions beyond the physical cuts. This severely restricts the allowed space by eliminating the R-matrices that have poles in that extended region. This is similar to removing R-matrices with bound states, but in this case, we can say that we remove resonances.1 We find that R-matrices that were not at vertices now appear at the vertices of the restricted space. In other cases, there is a one parameter family of integrable R-matrices. In that case, we find that all the boundary corresponds to integrable R-matrices. We also find vertices that do not seem to correspond to any known theory. The paper is organized as follows: in the rest of this section, we describe the properties of the S-matrix and R-matrix and exact results that follow from integrability with several examples. One result is new, we obtain an integrable reflection matrix for the periodic Yang-Baxter solution with no transmission (pYB). In the following section, we forget the requirement of integrability and just map out the allowed space of R-matrices from generic constraints. There we find the aforementioned properties. In the next section, we discuss the dual problem and find some useful properties of the problem with extended analyticity. In particular, we argue that it has to be regularized and that now, unitarity saturation does not follow automatically. In spite of that, all the R-matrices we found numerically actually saturate unitarity. In the last section, we give our conclusions. 1 Generically speaking, since some poles are too far from the real axis to be considered well-defined resonances. –2– JHEP04(2021)097 particular points of the space of allowed theories (or S-matrices) and that those particular points can be found by maximizing certain functionals in that space. For this paper, the case of interest is the 2d O(N ) non-linear sigma model studied in [7], exactly solved in [8, 9] and more recently revisited with the S-matrix bootstrap approach in [10–12]. In particular, in [10] it was argued that maximizing a linear functional in a convex space generically leads to vertices of the space. In fact, it was shown that the O(N ) non-linear sigma model (NLSM) lies at one of those vertices, the functional just being a way to find it. Later this was made more manifest in [13] where a section of the space was plotted with a clear (...truncated)