Spontaneous symmetry breaking in the $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ model for large $N$

We study the $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ generalization of the $\sigma$ model in the limit of large $N$, for four, three, two, and one space-time dimensions. We compute the effective potential and some momentum-dependent Green's functions. In one and two dimensions, spontaneous symmetry breakdown is impossible; any asymmetric minimum inserted in the tree-approximation potential is immediately filled in by the effects of radiative corrections. This is in agreement with general theorems. In four dimensions, the model is inconsistent; it possesses a tachyon. In three dimensions, the model seems to be consistent, and offers an interesting example of some nonlinear effects associated with spontaneous symmetry breakdown that are not present in the usual (tree-approximation) models.