Exchange Degeneracy andΣ−ΛSplitting

Abstract

The hypothesis is made that the sum of the imaginary parts of the baryonic Regge contributions to exotic meson-baryon scattering amplitudes must vanish. This hypothesis is combined with the assumptions that all coupling constants satisfy exact $\mathrm{SU}(3)$ symmetry and that the $\ensuremath{\Sigma}\ensuremath{-}\ensuremath{\Lambda}$ mass splitting cannot be neglected, so that the $\ensuremath{\Sigma}$ and $\ensuremath{\Lambda}$ lie on different Regge trajectories. All possible partitions of the spin-parity-${\frac{3}{2}}^{\ensuremath{-}}$ $\ensuremath{\Lambda}'\mathrm{s}$ and $\ensuremath{\Sigma}'\mathrm{s}$ (belonging to an octet, singlet, and decuplet) into the two trajectories are considered. A set of consistency equations is derived, and all solutions are obtained. If the $\ensuremath{\Lambda}(1520)$ is the only odd-parity hyperon on the bottom [$\ensuremath{\Lambda}(1115)$] trajectory, and the $\ensuremath{\Lambda}(1690)$ and one or two $\ensuremath{\Sigma}'\mathrm{s}$ lie on the top trajectory, there is only one solution. In this solution, there is no odd-parity decuplet, the $\frac{F}{D}$ ratios of the couplings to mesonbaryon states of the ${j}^{P}={\frac{1}{2}}^{+} \mathrm{and} {\frac{3}{2}}^{\ensuremath{-}}$ trajectories are 1, and the singlet-octet mixing angle of the ${\frac{3}{2}}^{\ensuremath{-}}$ $\ensuremath{\Lambda}'\mathrm{s}$ is $\ensuremath{\theta}=invtan\ensuremath{\surd}\frac{1}{2}$. This solution is in approximate agreement with experiment.