Symmetries and Nonsymmetries of the Relativistic Quark Model

Abstract

The dynamical successes of the relativistic quark model are separated from those due to symmetry arguments. Particular attention is paid to the example of the $q\overline{q}$, $L=1\mathrm{}$ meson decays. A relativistically invariant coplanar form of U(3) ⊗ U(3) is found to explain all predictions of the relativistic quark model for decays within a given multiplet except (a) the absence of spin-orbit effects and (b) relations between different partial waves, e.g., in $B\to \omega \pi$. The coplanar symmetry is expected to be as good as the quark-model classification scheme itself. It preserves most of the better predictions of $\mathrm{SU}{\mathrm{}\left(6\right)\mathrm{}}_W$, while avoiding some that are questionable. Some of the "good" predictions preserved are $\frac{\Gamma (A_2\to \rho \pi )}{\Gamma (A_2\to K\overline{K})}=6\mathrm{}$, ${\left(\frac{f}{d}\right)}_{8\left(\frac{5}{2^{-}}\right)\to 8\left(\frac{1}{2^{+}}\right)8\left(0^{-}\right)}=-\frac{1}{3}$, and ${\left(\frac{f}{d}\right)}_{8\left(\frac{5}{2^{+}}\right)\to 8\left(\frac{1}{2^{+}}\right)8\left(0^{-}\right)}=\frac{2}{3}$. Questionable predictions of $\mathrm{SU}{\mathrm{}\left(6\right)\mathrm{}}_W$ avoided by the coplanar symmetry include the selection rules $A_1\nrightarrow \rho (\lambda =0)\pi$, $B\nrightarrow \omega (\lambda =1)\pi$. A model of hadron decays which assumes them to occur via production of a ${}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}$ $q\overline{q}$ pair is consistent with (but more specific than) the coplanar symmetry.