Pion-Pion Scattering in aK-Matrix Model Incorporating Crossing Symmetry

A crossing-symmetric generalization of the $K$-matrix formalism is developed and used to construct a model for $\pi -\pi$ scattering. The full amplitude satisfies elastic unitarity in the $s$ channel and has the correct singularity structure at the elastic thresholds in the $t$ and $u$ channels. The two parameters present in the model are determined by requiring approximate satisfaction of the crossing relations in the neighborhood of the symmetry point. The resulting $p$-wave phase shift exhibits a resonance with mass about 800 MeV and width about 250 MeV. The $f^0$ is not reproduced. The $s$-wave scattering lengths in the $I=0\mathrm{}$ and $I=2\mathrm{}$ channels are $(a_0,a_2)=(-0.67,-0.30\mathrm{})\mathrm{}$ and the effective Chew-Mandelstam coupling constant is found to be 0.18.