Pion-Pion Dynamics in theσModel

We propose an unambiguous way of constructing amplitudes which satisfy both unitarity and the current-algebra constraints. This consists in working out higher-order corrections on a Lagrangian which produces the correct soft-pion limit in the tree approximation. We consider $\pi \pi$ scattering in the $\sigma$ model, and we compute the perturbation series up to second order. The renormalization procedure preserves the partially conserved axial-vector current condition and the current-algebra constraints at each order. In order to sum the strong-coupling perturbation series, we use the Padé-approximation technique. Thereby, our partial-wave amplitudes satisfy unitarity. The $\rho$ and $f_0$ resonances are generated, although they were not present in the Lagrangian. Our unitary amplitudes satisfy crossing symmetry to a very good accuracy, showing the consistency of the results. Our results are in agreement with the "up-down" solution of the $I=0\mathrm{}$, $s$-wave $\pi \pi$ phase shift, with a very broad $\sigma$ resonance; the $I=2\mathrm{}$ $s$-wave phase shift is repulsive, and agrees very well with experiment.