Strong-Interaction Symmetries Based upon Rank-Three Lie Groups

We consider all rank-three simple Lie groups as possible candidates for a higher symmetry of strong interactions. All such groups imply the existence of a new quantum number $X$, the oddness, and of odd particles with nonzero values of $X$. Because of uncertainties in the experimental observation of these particles, we look for evidence of such symmetries in the properties of ordinary ($X=0\mathrm{}$) particles. We give arguments to show that $\phi$ decay into $\rho$ and $\pi$ mesons is a particularly good place to look for such evidence. In all groups, we assign the vector mesons to the regular representation and derive mass formulas and decay rates for various assignments of the pseudoscalar mesons and the mass operator to representations of the group. We find that it is possible to formulate a general criterion, which can be applied to all rank-three Lie groups, for assigning these representations, and that with this assignment all such groups give the same mass formula and decay widths for the vector mesons, namely, $(3\mathrm{}\omega +\rho -4\mathrm{}{K}^{*})(3\mathrm{}\phi +\rho -4\mathrm{}{K}^{*})+8\mathrm{}{(\rho -K^{*})}^2=0\mathrm{}$ and $\Gamma (\phi \to \rho \pi )=0.3-0.6\mathrm{} \mathrm{MeV}, \Gamma (\phi \to K\overline{K})=2\mathrm{} \mathrm{MeV},$ in extremely good agreement with experiment. We summarize the main properties of rank-three Lie groups in appendices.