Quantum Integrability and Exact S-Matrices for Affine Toda Theories

Abstract

The study of perturbed two-dimensional conformal theories has attracted recently much attention. Following the suggestion of A.B. Zamolodchikov [1] it has become clear that a way to obtain information about the off-critical theories is to apply perturbation theory around their ultraviolet fixed points. In general the perturbation of a conformal field theory will induce a renormalization group flow from the initial (ultraviolet) fixed point to either a) a new (infrared) fixed point where the model is again conformal invariant (Zamolodchikov c-theorem [2]), or, b) a flow to a system with finite correlation length. In particular it has been suggested that for certain classes of perturbations some of these models may possess an infinite number of conserved currents and therefore retain the integrability properties they have at their fixed, conformal points. In more than two dimensions the Coleman-Mandula theorem [3] forces a theory with higher-spin conserved currents to have a trivial S-matrix. In two dimensions this is not true, however the presence of higher-spin charges which commute with the S-matrix has a profound effect on the structure of the scattering amplitudes of these theories: the n-particle S-matrices factorize into a product of elastic two-particle S-matrices which can be determined exactly using unitarity and a bootstrap principle [4]. Therefore in these cases, by studying perturbed conformal field theories one can find all the on-shell informations of the massive theory which are encoded in the S-matrix. The integrals of motion and the S-matrices determined in the conformal field theory approach were recognized as characteristic of a class of two-dimensional field theories known as Toda theories [5]. The basic properties of a Toda system stem from an underlying Lie algebraic construction and depending on whether the algebra is affine or not, the resulting theory is massive or conformally invariant respectively. Thus an affine Toda theory obtained from the corresponding Toda theory by affinizing the Lie algebra can be interpreted as the integrable deformation of a conformal field theory.