Coupled Channel Approach toJ=32+Resonances in the Unitary Symmetry Model

An analysis is made of the $p_{\frac{3}{2}}$ pseudoscalar meson-baryon scattering amplitudes in all isotopic spin and strangeness states. In analogy with pion-nucleon scattering, the basic assumption that the single baryon exchange contribution provides the dominant force in these amplitudes is made, but all the coupled two-particle channels in each isotopic spin and strangeness state are included. A relativistic scattering matrix for each state is constructed by a matrix formulation of the $\frac{N}{D}$ method so that it satisfies the requirements of unitarity and symmetry. Being fully relativistic, unlike the static model, the theory does not contain adjustable cutoff parameters. The known masses of the mesons and baryons and the Yukawa-type meson-baryon coupling constants are the only parameters that enter into the calculation. The octet model of Gell-Mann and Ne'eman is used to define the coupling constants in terms of a single parameter $f$ and the known pion-nucleon coupling constant. Then the requirement that $N^{*}(T=\frac{3}{2}; S=0)$, ${Y_1}^{*}(T=1; S=-1)$, the recently observed ${\Xi }^{*}(T=\frac{1}{2}; S=-2)$, and a yet-to-be-discovered $Z^{-}(T=0, S=-3)$ exist as a tenfold representation of the unitary symmetry model restricts $f$ to a rather narrow range. There are no other resonances in the $p_{\frac{3}{2}}$ state for $f$ in this range, except possibly one in the $T=0\mathrm{}$, $S=-1\mathrm{}$ system. The positions and widths of these resonances are discussed.