The theory of non-renormalizable interactions: The large N expansion

A particular class of non-renormalizable interactions is studied in the infinite cut-off limit. In this paper we consider the quadrilinear interaction of an N-component field; the Lagrangian is invariant under the action of the O(N) group. The Green functions are expanded in powers of 1/N; we prove that this expansion is finite and renormalizable at all orders in not too high dimensions, the outputs are not C^∞ in the coupling constant around the origin: this property explains why divergences are present in the standard perturbative expansion. The interactions of both spin-zero and $\text{spin}\text{-}\text{1}\text{2}$ fields have been studied: peculiar problems arise in the case of a current-current interaction.