|ΔI|=12Rule From the Symmetric Quark Model

Two topics are treated in this paper: the explanation of the $\left|\Delta I\right|=\frac{1}{2}$ rule based on the symmetric quark model and the tests for this explanation in ${\Omega }^{-}$ nonleptonic decays. From the "color-quark," the three-triplet, and the paraquark models, the $\left|\Delta I\right|=\frac{1}{2}$ rule follows for the octet-hyperon, the ${\Omega }^{-}$, and the kaon weak nonleptonic decays as a consequence of current algebra, pion PCAC (partially conserved axial-vector current), and dispersion relations. Gronau's successful numerical results for the octet-hyperon decay amplitudes also follow in these alternatives to the Bose-quark model. However, though the origin of both explanations is the Fierz reshuffling property of the $V\pm A$ interactions, the explanations of the $\left|\Delta I\right|=\frac{1}{2}$ rule are quite distinct, e.g., in the Bose-quark model this rule is exact whereas in the other versions it is only approximate, being violated by continuum contributions to the absorptive parts. Because $〈0\left|{\mathcal{H}}_w\right|K〉$ and $〈\pi \left|{\mathcal{H}}_w\right|K〉$ vanish in the symmetric quark model, the usual current-algebra soft-pion argument for $\left|\Delta I\right|=\frac{1}{2}$ rule and the $K^{*}$-pole-dominance assumption (as a Feynman diagram) for $K_0^1\to 2\pi$ are not convincing. On the other hand, the ordinary Fermi-quark model supplemented with octet dominance can be excluded, as it predicts $\frac{D}{F}=3\mathrm{}$ in the SU(3) limit for the matrix element of the parity-conserving Hamiltonian for two baryons in the nucleon octet $\frac{D}{F}\simeq -0.85\mathrm{}$ from $P$-wave fits). The $\Lambda K^{-}$ decay mode of the ${\Omega }^{-}$ should be predominantly $P$ wave (parity-conserving), whereas the $\Xi \pi$ mode should have the $P$ wave strongly suppressed and comparable to the $D$ wave (parity-violating). This implies $\frac{\Gamma ({\Omega }^{-}\to \Xi \pi )}{\Gamma ({\Omega }^{-}\to \Lambda K^{-})}\ll 1$. The estimated total ${\Omega }^{-}$ decay rate is consistent with the present experimental number.