Pion Mass Effects in the Large $N$ Limit of $\chiPT$

Abstract

We compute the large $N$ effective action of the $O(N+1)/O(N)$ non-linear sigma model including the effect of the pion mass to order $m^2_π/f_π^2$. This action is more complex than the one corresponding to the chiral limit not only because of the pion propagators but also because chiral symmetry produce new interactions proportional to $m^2_π/f_π^2$. We renormalize the action by including the appropriate counter terms and find the renormalization group equations for the corresponding couplings. Then we estudy the unitarity propierties of the scattering amplitudes. Finally our results are applied to the particular case of the linear sigma model and also are used to fit the pion scattering phase shifts.