O( N) invariant quantum field theoretical models: Exact solution

Abstract

The development of the theory of quantum integrable systems has led to solutions of quantum field theory models in 1 + 1 dimensions [1–3]. The most interesting models among them are those with asymptotic freedom and dimensional transmutation [3], e.g. the chiral Gross-Neveu model, the isotopic Thirring model and its anisotropic generalizations. Before an exact solution of these models was found, some results were obtained by traditional methods. For example, the mass spectrum and S-matrices of excitations were obtained from the phenomenological theory of the factorizable S-matrix [4–7]. Renormalization group equations for the invariant charge were obtained via perturbation theory. The same results were obtained by means of the Bethe ansatz method in [3,8] for the SU( n) symmetric and Z 2 symmetrical models. Assuming that the S-matrices for O(2 k) Gross-Neveu model are factorizable one is able to calculate the mass spectrum and the S-matrices for this model exactly [6,7]. It remained however to verify this crucial assumption and to calculate the S-matrices by means of the Bethe ansatz method. In the present work we give an exact solution of the relativistic O(2 k) symmetrical model with a four-fermion interaction by means of the Bethe ansatz. The mass spectrum and the S-matrices in this model are calculated. The connection between this model and the O(2 k) Gross-Neveu model is discussed.