Nonleptonic decays in a renormalizable gauge model of chiral symmetry

Abstract

We present a model based on chiral SU(3) \ifmmode\times\else\texttimes\fi{} SU(3) for the nonleptonic decays of mesons and baryons. The model is fully gauge-invariant and renormalizable. The basic fields in the model are the spin-1 and spin-0 mesons. All symmetry-breaking effects, including the nonleptonic weak mixings, are achieved through the spontaneous-symmetry-breaking mechanism. Our choice of the Higgs-Kibble scalars automatically ensures octet dominance for the nonleptonic weak vertex. Since we find it hard to include baryons in the model, a phenomenological treatment of the baryon decays, assuming SU(3) invariance of the baryonic couplings, is presented. In the model we calculate the two-pion and three-pion decays of the kaon, the $s$-wave amplitudes for the hyperon decays, and the ${K}_{L}\ensuremath{-}{K}_{S}$ mass difference. The results for $K\ensuremath{\rightarrow}2\ensuremath{\pi}$ and $K\ensuremath{\rightarrow}3\ensuremath{\pi}$ decay widths are in excellent agreement with experiment. The slope parameter for the $K\ensuremath{\rightarrow}3\ensuremath{\pi}$ decay, however, comes out with the wrong sign. The $s$-wave amplitudes for the hyperon decays agree reasonably well with experiment. The estimate for the ${K}_{L}\ensuremath{-}{K}_{S}$ mass difference is of the correct order of magnitude. ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{0}$ decay is calculated by using the known current-algebra estimate of $\ensuremath{\eta}{\ensuremath{\pi}}^{0}$ mixing. The decay width obtained by us is rather low. Our conclusion is that $\ensuremath{\eta}{\ensuremath{\pi}}^{0}$ mixing alone is not sufficient to explain the ${K}^{+}$ decay. On extending the model to chiral SU(4) \ifmmode\times\else\texttimes\fi{} SU(4), we predict the existence of a component transforming as the 15 representation of SU(4) in the nonleptonic Hamiltonian. Therefore, the nonleptonic decays of charmed mesons will provide a definitive test for the model.