Pion-Pion Dynamics in theσModel

Abstract

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 $\ensuremath{\pi}\ensuremath{\pi}$ scattering in the $\ensuremath{\sigma}$ 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 Pad\'e-approximation technique. Thereby, our partial-wave amplitudes satisfy unitarity. The $\ensuremath{\rho}$ 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 $I=0$, $s$-wave $\ensuremath{\pi}\ensuremath{\pi}$ phase shift, with a very broad $\ensuremath{\sigma}$ resonance; the $I=2$ $s$-wave phase shift is repulsive, and agrees very well with experiment.