Sum Rules for Mass Differences Within the Baryon Decuplet and the Meson Nonets

A group-theoretic approach, previously used to derive a precise hybrid formula relating electromagnetic mass differences of pseudoscalar mesons and baryons, is extended here to the barycn decuplet and the vector- and tensor-meson nonets. We derive three sum rules relating mass differences within isotopic-spin multiplets of the decuplet, as well as a hybrid formula relating baryon-octet and -decuplet electromagnetic mass differences. Using the present experimental average for the ${\Xi }^{*-}-{\Xi }^{*0\mathrm{}}$ mass difference, the hybrid formula can be used to predict $M_{Y^{-}}-M_{Y^{+}}=3.2\mathrm{}\pm 0.5$ MeV, $M_{{\Delta }^0}-M_{{\Delta }^{+}}=-0.24\mathrm{}\pm 0.16$ MeV, and $M_{{\Delta }^{++}}-M_{{\Delta }^{-}}=0.72\mathrm{}\pm 0.48$ MeV. No assumptions concerning $U$-spin symmetry are required in obtaining these results. With the 27-dimensional part of the SU(2)-breaking interaction taken to be a $U$-spin singlet, two additional predictions follow: $M_{Y^{-}}-M_{Y^0}=3.4\mathrm{}\pm 0.6$ MeV and $M_{{\Delta }^{++}}-M_{{\Delta }^{+}}=3.9\mathrm{}\pm 0.9$ MeV. Hybrid mass formulas are also derived for meson nonets. Using the present experimental average for the $K^{{*}_0}-K^{{*}_{+}}$ mass difference, we predict the neutral $\rho$ meson to be heavier than its charged counterparts by approximately 5 ± 2 MeV.