Baryon Mass Spectra and Baryon Couplings

General intermultiplet mass formulas between the decuplet and octet baryons are derived, based on the chiral $\mathrm{SU}\left(3\right)\mathrm{}\otimes \mathrm{SU}\left(3\right)\mathrm{}$ charge algebra and the asymptotic $\mathrm{SU}\left(3\right)\mathrm{}$ symmetry imposed only on the charge operator $V_K$ which is the $\mathrm{SU}\left(3\right)\mathrm{}$ raising or lowering operator in the symmetry limit. In the absence of particle mixing, the formulas take on a simple form and are useful as a first guide in deducing baryon mass spectra. Inclusion of particle mixing is possible. The formulas imply that the equal squared-mass spacings of decuplet states ${\left({\delta }_a\right)}^2$ ($a$ specifies the quantum numbers of the $a$ decuplet) are universal [i.e., ${\left({\delta }_a\right)}^2=\mathrm{const}$] and, furthermore, equal to the universal spacings of the $\Xi$ and $\Sigma$ members of any octet baryons. In the case of the usual ${\mathrm{½}}^{+}$ and ${\frac{3}{2}}^{+}$ baryons, the formulas coincide with the well-known $\mathrm{SU}\left(6\right)\mathrm{}$ mass formula. Broken-$\mathrm{SU}\left(3\right)\mathrm{}$ sum rules (in the absence of mixing) for general transitions of the form $\left(\mathrm{octet}\mathrm{baryon}\right)\to \left(\mathrm{octet}\mathrm{baryon}\right)+\pi (or l+\overline{\nu })$ are also obtained. In particular, the strong decays, ${\frac{5}{2}}^{-}\to {\frac{1}{2}}^{+}+\pi$ and ${\frac{5}{2}}^{+}\to {\frac{1}{2}}^{+}+\pi$, are discussed in detail. The sum rules, in general, give rise to significant $\mathrm{SU}\left(3\right)\mathrm{}$-breaking effects. However, for the familiar axial-vector ${\frac{1}{2}}^{+}\to {\frac{1}{2}}^{+}+l+\overline{\nu }$ transitions, our broken-$\mathrm{SU}\left(3\right)\mathrm{}$ sum rules assume the same forms as satisfied by the hypothetical exact $\mathrm{SU}\left(3\right)\mathrm{}$ couplings. This justifies the use of the original Cabibbo analysis [in broken-$\mathrm{SU}\left(3\right)\mathrm{}$ symmetry] in determining the value of the axial-vector Cabibbo angle.