SU(6)Algebra of Gell-Mann: Mass Splittings and Nonleptonic Decays of Hadrons

Abstract

We apply the $\mathrm{SU}\left(6\right)\mathrm{}$ algebra of Gell-Mann to nonleptonic decays of hadrons. We assume that the weak-interaction Hamiltonian for $s$-wave decays transforms like $S_7^{}{}_{}{}^5$ (the seventh component of the space integral of the pseudoscalar quark density). Using the Fubini-Furlan-Adler-Weisberger technique and the partially conserved axial-vector current, we express the $s$-wave decay amplitudes in terms of the reduced matrix elements, which are the same as the ones that appear in the expression for mass splittings if one assumes that the symmetry-breaking part of the mass operator transforms like $S_8$ (the eighth component of the space integral of the scalar quark density). Making use of the fact that the mass-splitting parameter is universal, and assuming that the $s$-wave coupling constant $G_s$ is also universal, we predict a ratio $\frac{(K_1^{}{}_{}{}^0\to 2\pi )}{A\left(\Sigma _{-}^{}{}_{}{}^{-}\right)}$ which is in excellent agreement with experiment. Knowing any one of the hyperon $s$-way decay amplitudes we can predict the others. Also we get $A\left(\Sigma _{+}^{}{}_{}{}^{+}\right)=0\mathrm{}$. The implications of these results are discussed.