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O(N)-symmetric vector models for N large

So far, universal properties of O(N) symmetric critical systems have been derived within the framework of the formal ϵ = 4−d expansion. Therefore, it is reassuring to verify that the results obtained in this way remain valid, at least in some limiting case, even when ϵ is not infinitesimal. Here it is shown in the example of the O(N) symmetric (ϕ2)2 field theory, the same universal properties can also be derived at fixed dimension in the large N limit and, more generally, order by order in an 1/N expansion. Moreover, large N techniques are also useful, because they provide an insight into other non-perturbative questions, including issues relevant to four-dimensional physics like renormalons and triviality. Using a large N expansion, we exhibit a remarkable relation between the (ϕ2)2 field theory and the non-linear σ-model, valid to all orders.