Low-Energyπ−ΛInteraction, and theΣΛπCoupling Constant

A semiphenomenological study of the dynamics of the first hyperon resonance ${Y_1}^{*} \mathrm{}\left(1385\right)\mathrm{}$ has been made in a one-channel approximation by the techniques of partial-wave dispersion relations. Contributions from $\Sigma$ and ${Y_1}^{*}$ exchange, and the exchange of a low-energy $s$-wave $\pi -\pi$ pair, to dispersion relations for the $P_{\frac{3}{2}}$ $\pi -\Lambda$ scattering amplitude were evaluated on the physical and crossed physical cuts. The coupling constant $g\Sigma \Lambda \pi$ and a parameter related to the $P_{\frac{3}{2}}\pi -\Lambda$ scattering length were varied so that the amplitude calculated from the dispersion relation was consistent with the input resonant amplitude on both cuts simultaneously. This could be achieved for $g\Sigma \Lambda {\pi }^2=10.9\mathrm{}\pm 0.3$ The dynamics of ${Y_1}^{*}$ are found to depend almost as much on the exchange of the $s$-wave $\pi -\pi$ pair as on $\Sigma$ exchange.