Relation among theΞ12*,Y1*,N32*Decay Widths in Broken Unitary Symmetry

The decay amplitudes of the decuplet particles ${{\Xi }_{\frac{1}{2}}}^{*}\mathrm{}\left(1530\right)\mathrm{}$, ${Y_1}^{*}\mathrm{}\left(1385\right)\mathrm{}$, and ${N_{\frac{3}{2}}}^{*}\mathrm{}\left(1238\right)\mathrm{}$ into a baryon and a pion have been calculated assuming that the interaction giving rise to such decays consists of a unitary symmetric part plus all the terms violating unitary symmetry which transform as the ${T_3}^3$ component of a unitary tensor. It is shown that contrary to what one would expect at first sight, only three undetermined coupling constants appear in the calculation of the four amplitudes $G_1$, $G_2$, $G_3$, $G_4$ for the decays (1) ${{\Xi }_{\frac{1}{2}}}^{*}\to \Xi +\pi$, (2) ${Y_1}^{*}\to \Sigma +\pi$, (3) ${Y_1}^{*}\to \Lambda +\pi$, (4) ${N_{\frac{3}{2}}}^{*}\to N+\pi$. One can therefore predict a relation among the above four amplitudes, which turns out to be: $G_4=-\left(\frac{3}{\sqrt{2}}\right)G_3-\left(\surd \frac{3}{2}\right)G_2+\left(\surd 6\right)G_1.$ With the present values of the widths this relation is satisfied remarkably well.