One-Loop Renormalization of the NonlinearσModel

We propose a method for regularizing and renormalizing the nonlinear $\sigma$ model based on the limit of the linear $\sigma$ model when the $\sigma$ mass tends to infinity. The linear $\sigma$ model being renormalized using Ward identities, this procedure ensures automatically the soft-pion theorems. It also retains the properties of transformations of the Green's functions under the chiral group. In the one-loop approximation, we show that the pion propagator has a finite limit when the $\sigma$ mass tends to infinity, while the four-pion Green's function diverges only logarithmically with the $\sigma$ mass.