The two - dimensional O ( 3 ) nonlinear σ - model at finite temperature

We present a direct derivation of the thermodynamic integral equations of the O(3) nonlinear σ-model in two dimensions. January 1992, Phys.Lett.B, in press on leave from Departamento de Fisica, Universidade Federal de S.Carlos, C.P. 676 S.Carlos 13560, Brazil. [email protected] The two-dimensional O(3) nonlinear σ-model is a well known example of an exactly integrable field theory [2, 3] which is asymptotically free at short distances and a mass gap appears in the spectrum through the dimensional transmutation mechanism [1]. Its action A is defined by A = 1 g ∫ dx(∂ν~η) 2 (1) where ~η = (nx, ny, nz) is a three component unit vector on the sphere S ; ~η = 1. It is believed that the O(3) nonlinear σ-model has no bound states and its fundamental particles belong to the O(3) multiplet. Using these hypotheses and the factorized bootstrap A. Zamolodchikov and Al. Zamolodchikov [4] have found the respective two-body S-matrices S ij (β); S ij (β) = π + iβ (2π + iβ)(π − iβ) ( 2πiβ π + iβ δi,jδk,l − 2πδk,jδi,l − iβδk,iδl,j ) (2) The solution for the O(3) nonlinear σ-model directly from its bosonic representation (1) is still unknown. However, Wiegmann [6], using a fermionic formulation proposed early by Polyakov and Wiegmann [5], was able to find the associated Bethe ansatz equations from a subtle limit of an exactly solved four fermion model. These equations are given by, ei = N ∏ j=1,i6=j βi − βj + iπ βi − βj − iπ M ∏ j=1 βi − λj + iπ βi − λj − iπ , i = 1, 2, ..., N (3)