1/N expansion of the nonlinear σ model and its renormalization through stochastic quantization

The 1/N expansion of the nonlinear σ model is considered in the framework of Parisi-Wu stochastic quantization. The expansion is implemented both in the Langevin approach and also in the functional-integral representation of stochastic processes. Whereas in the first method quantization is possible only by means of an artifice, the functional-integral approach is straightforward and, thanks to a graphical Ward identity and the Becchi-Rouet-Stora-Tyutin symmetry characteristic of the formulation, we show the renormalizability of the model in two and three dimensions