$\chi_{b}(3P)$ Multiplet Revisited: Ultrafine Mass Splitting and Radiative Transitions

Invoked by the recent CMS observation regarding candidates of the $\chi_b(3P)$ multiplet, we analyze the ultrafine and mass splittings among $3P$ multiplet in our unquenched quark model (UQM) studies. The mass difference of $\chi_{b2}$ and $\chi_{b1}$ in $3P$ multiplet measured by CMS collaboration ($10.6 \pm 0.64 \pm 0.17$ MeV) is very close to our theoretical prediction ($12$ MeV). Our corresponding mass splitting of $\chi_{b1}$ and $\chi_{b0}$ enables us to predict more precisely the mass of $\chi_{b0}(3P)$ to be ($10490\pm 3$) MeV. Moreover, we predict ratios of the radiative decays of $\chi_{bJ}(nP)$ candidates, both in UQM and quark potential model. Our predicted relative branching fraction of $\chi_{b0}(3P)\to\Upsilon(3S)\gamma$ is one order of magnitude smaller than $\chi_{b2}(3P)$, this naturally explains the non-observation of $\chi_{b0}(3P)$ in recent CMS search. We hope these results might provide useful references for forthcoming experimental searches.