On String S-matrix, Bound States and TBA

TL;DR

This paper analyzes the S-matrix and bound states in the mirror model of the light-cone AdS superstring, revealing the uniqueness of the S-matrix under unitarity and exploring the complex structure of bound states and their finite J corrections.

Contribution

It demonstrates the essentially unique form of the mirror S-matrix satisfying unitarity and explores the structure and solutions of bound state equations in the AdS superstring context.

Findings

The mirror S-matrix is fixed by unitarity and is a meromorphic function on an elliptic curve.

Multiple solutions exist for the bound state equations, with some exhibiting pathological behaviors at finite J.

Most bound state solutions show signs of complex energy corrections or deviations from the asymptotic Bethe ansatz at large but finite J.

Abstract

The study of finite J effects for the light-cone AdS superstring by means of the Thermodynamic Bethe Ansatz requires an understanding of a companion 2d theory which we call the mirror model. It is obtained from the original string model by the double Wick rotation. The S-matrices describing the scattering of physical excitations in the string and mirror models are related to each other by an analytic continuation. We show that the unitarity requirement for the mirror S-matrix fixes the S-matrices of both theories essentially uniquely. The resulting string S-matrix S(z_1,z_2) satisfies the generalized unitarity condition and, up to a scalar factor, is a meromorphic function on the elliptic curve associated to each variable z. The double Wick rotation is then accomplished by shifting the variables z by quarter of the imaginary period of the torus. We discuss the apparent bound states of the string and mirror models, and show that depending on a choice of the physical region there are one, two or 2^{M-1} solutions of the M-particle bound state equations sharing the same conserved charges. For very large but finite values of J, most of these solutions, however, exhibit various signs of pathological behavior. In particular, they might receive a finite J correction to their energy which is complex, or the energy correction might exceed corrections arising due to finite J modifications of the Bethe equations thus making the asymptotic Bethe ansatz inapplicable.