Some exact results for the O( N)-symmetric non-linear σ-model to O(1/ N)

Abstract

With the non-linear σ-model on the lattice in view, we study its continuum version with a finite, sharp momentum cut-off. We find exact analytical results for its mass gap and magnetic susceptibility to two leading orders in 1/ N. Asymptotically, at weak coupling, our results converge to those of renormalization group improved weak coupling perturbation theory to two-loop order, and give the ratio of the mass to the Λ-parameter analytically. Thus our results provide an opportunity to study the rate at which asymptotic scaling is approached. Since this approach is slow for low values of N, we find a region of intermediate coupling strenghts, where our results describe non-asymptotic scaling of a well-defined and universally valid kind.