We compute the large N effective action of the O(N+1)/O(N) nonlinear σ model including the effect of the pion mass to order ${\mathit{m}}_{\mathrm{\pi }}^2$/${\mathit{f}}_{\mathrm{\pi }}^2$. This action is more involved than the one corresponding to the chiral limit not only due to the pion propagators but also because the chiral symmetry produces new interactions proportional to ${\mathit{m}}_{\mathrm{\pi }}^2$/${\mathit{f}}_{\mathrm{\pi }}^2$. We renormalize the action by including the appropriate infinite set of counterterms and we obtain the renormalization-group equations for the corresponding couplings. Then we study the unitarity properties of the scattering amplitudes. Finally our results are applied to the particular case of the linear σ model and also are used to fit the pion scattering phase shifts. In spite of the fact that the model has an infinite number of parameters to fit and, therefore, it does not have predictive power in a strict sense, a particularly simple choice of the couplings produces a good fit of the J=0 phase shifts.

• Received 22 July 1994

DOI:https://doi.org/10.1103/PhysRevD.52.2878