On the Vacuum Energy Density and Non-Perturbative Effects in Sigma   Models
  The vacuum energy density is obtained for the $O(N)$ nonlinear sigma model. It is shown that non-perturbative contributions are connected with the square of the symmetry current of the group $O(N)$. This result is valid for $\sigma$- fields which are subject to the constraint. 
V.G.Ksenzov