Dynamics of thed32Unitary Multiplets

A dynamical model of pseudoscalar meson-baryon scattering in the $J^P=\frac{3}{2^{-}}$ state is proposed to support the conjecture of Glashow and Rosenfeld that the $N^{**}\mathrm{}\left(1512\right)\mathrm{}$ resonance is a member of a unitary-symmetry octet. The dynamical mechanism analyzed here is based on the Cook-Lee model of the higher pion-nucleon resonances. It is shown that the coupling to inelastic vector meson-baryon states in a generalized two-channel formalism yields an octet of $d_{\frac{3}{2}}$ resonances and a unitary singlet as well. The baryon, pseudoscalar-meson, and vector-meson octets are each assumed degenerate, so that the Cook-Lee model is immediately adaptable to an analysis of resonant unitary multiplets as a function of $f$, the Yukawa mixing parameter. It is found that, for $f=0.326\mathrm{}$, the attraction is slightly greater for the octet than for the singlet. The 2×2 octet amplitude is diagonalized by a rotation through an angle ${\theta }^{*}=45\mathrm{}°$; the Yukawa mixing parameter is $f^{*}=0.428\mathrm{}$. The $N^{**}N\pi$ coupling constant is computed to be $\frac{g^{*2\mathrm{}}}{\left(4\mathrm{}\pi \right)}=0.150\mathrm{} {m_{\pi }}^{-2\mathrm{}}$, which may be compared with the observed $N^{**}$ width.