|ΔI|=12Rule From the Symmetric Quark Model

Abstract

Two topics are treated in this paper: the explanation of the $|\ensuremath{\Delta}I|=\frac{1}{2}$ rule based on the symmetric quark model and the tests for this explanation in ${\ensuremath{\Omega}}^{\ensuremath{-}}$ nonleptonic decays. From the "color-quark," the three-triplet, and the paraquark models, the $|\ensuremath{\Delta}I|=\frac{1}{2}$ rule follows for the octet-hyperon, the ${\ensuremath{\Omega}}^{\ensuremath{-}}$, 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 $V\ifmmode\pm\else\textpm\fi{}A$ interactions, the explanations of the $|\ensuremath{\Delta}I|=\frac{1}{2}$ 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 $〈0|{\mathcal{H}}_{w}|K〉$ and $〈\ensuremath{\pi}|{\mathcal{H}}_{w}|K〉$ vanish in the symmetric quark model, the usual current-algebra soft-pion argument for $|\ensuremath{\Delta}I|=\frac{1}{2}$ rule and the ${K}^{*}$-pole-dominance assumption (as a Feynman diagram) for ${K}_{0}^{1}\ensuremath{\rightarrow}2\ensuremath{\pi}$ are not convincing. On the other hand, the ordinary Fermi-quark model supplemented with octet dominance can be excluded, as it predicts $\frac{D}{F}=3$ in the SU(3) limit for the matrix element of the parity-conserving Hamiltonian for two baryons in the nucleon octet $\frac{D}{F}\ensuremath{\simeq}\ensuremath{-}0.85$ from $P$-wave fits). The $\ensuremath{\Lambda}{K}^{\ensuremath{-}}$ decay mode of the ${\ensuremath{\Omega}}^{\ensuremath{-}}$ should be predominantly $P$ wave (parity-conserving), whereas the $\ensuremath{\Xi}\ensuremath{\pi}$ mode should have the $P$ wave strongly suppressed and comparable to the $D$ wave (parity-violating). This implies $\frac{\ensuremath{\Gamma}({\ensuremath{\Omega}}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\Xi}\ensuremath{\pi})}{\ensuremath{\Gamma}({\ensuremath{\Omega}}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\Lambda}{K}^{\ensuremath{-}})}\ensuremath{\ll}1$. The estimated total ${\ensuremath{\Omega}}^{\ensuremath{-}}$ decay rate is consistent with the present experimental number.