Current Algebra ofSU(6)Wand Nonleptonic Hyperon Decays

The parity-violating and parity-conserving parts of the nonleptonic hyperon decay amplitudes are studied in the framework of the group ${\mathrm{SU}\left(6\right)\mathrm{}}_W$. Describing the decay in terms of the usual weak Hamiltonian $H_w$ bilinear in Cabibo currents by the amplitude $〈B_{2}M\left|H_w\mathrm{}\right(0\left)\right|B_1〉$, the meson $M$ is contracted from the final state, the partially conserved axial-vector-current hypothesis is introduced and the meson is taken off the mass shell. The amplitude is then evaluated at an arbitrary point off the mass shell of the meson, assuming the quark model for the currents and a single-particle pole approximation. The nine $S$-wave and twelve $P$-wave decays of the baryons in the 56-dimensional representation of ${\mathrm{SU}\left(6\right)\mathrm{}}_W$ are found to be determined by four parameters, one of which relates the $S$- and $P$-wave decays. A new $P$-wave sum rule for the baryon octet decays $8\sqrt{3}P\left({{\Xi }_{-}}^{-}\right)=3P\left({{\Sigma }_0}^{+}\right)+3\mathrm{}\sqrt{3}P\left({{\Lambda }_{-}}^0\right)-\left(\frac{5}{\sqrt{2}}\right)P\left({{\Sigma }_{+}}^{+}\right)$ is found, as well as new sum rules relating the ${\Omega }^{-}$ decay to the decays of the baryon octet. With the latter result, the nonleptonic decay rates of the ${\Omega }^{-}$ are calculated from experimental octet decay amplitudes.