Sum Rules for Coupling Constants in Exact and BrokenSU(6)W

The meson-baryon vertex and the decuplet decays are studied within the framework of the exact and the broken $\mathrm{SU}{\mathrm{}\left(6\right)\mathrm{}}_W$ group. The breaking is provided by a $W$-spin scalar spurion transforming like the $I=0\mathrm{}$, $Y=0\mathrm{}$ member of the 35-dimensional adjoint representation of $\mathrm{SU}{\mathrm{}\left(6\right)\mathrm{}}_W$. Twenty-one sum rules are obtained in the broken symmetry. One of these directly relates the ($p,p{\pi }^0$) and ($N^{*++},p{\pi }^{+}$) couplings and yields a decay width $\Gamma (N^{*++}\to p{\pi }^{+})=70\mathrm{}$ MeV. Another relates the vertices ($Y_1^{}{}_{}{}^{*+},\Lambda {\pi }^{+}$) and ($Y_1^{}{}_{}{}^{*+},{\Sigma }^{+}{\pi }^0$), which seems to agree with the latest experiments on $Y_1^{}{}_{}{}^{*}$ decays. One more relation among the observable decuplet decays is the Gupta-Singh sum rule, which is very well satisfied experimentally. Some predictions of exact $\mathrm{SU}{\mathrm{}\left(6\right)\mathrm{}}_W$ seem to disagree with experiment. It is concluded that in the context of three-point functions the predictions of the broken $\mathrm{SU}{\mathrm{}\left(6\right)\mathrm{}}_W$ group are in very good agreement with experiments.