Renormalized effective actions for the O(N) model at next-to-leading order of the 1/N expansion

A fully explicit renormalized quantum action functional is constructed for the O(N) model in the auxiliary field formulation at next-to-leading order (NLO) of the 1/N expansion. Counterterms are consistently and explicitly derived for arbitrary constant vacuum expectation value of the scalar and auxiliary fields. The renormalized NLO pion propagator is exact at this order and satisfies Goldstone's theorem. Elimination of the auxiliary field sector at the level of the functional provides with O(N0) accuracy the renormalized effective action of the model in terms of the original variables. Alternative elimination of the pion and sigma propagators provides the renormalized NLO effective potential for the expectation values of the N vector and of the auxiliary field with the same accuracy.