Dominant spin-orbit effects in radiative decays {$\Upsilon(3S\rightarrow \gamma\chi_{bJ}(1P))$}}

Abstract

We show that there are two reasons why the partial width for the transition $\Gamma_1(\Upsilon(3S)\rightarrow \gamma\chi_{b1}(1P))$ is suppressed. Firstly, the spin-averaged matrix element (m.e.) $\bar{I(3S|r|1P_J)}$ is small, being equal to 0.023 GeV$^{-1}$ in our relativistic calculations. Secondly, the spin-orbit splittings produce relatively large contributions, giving $I(3S|r|1P_2)=0.066$ GeV$^{-1}$, while due to large cancellation the m.e. $I(3S|r|1P_1)=-0.020$ GeV$^{-1}$ is small and negative; at the same time the magnitude of $I(3S|r|1P_0)=-0.063$ GeV$^{-1}$ is relatively large. These m.e. give rise to the partial widths: $\Gamma_2(\Upsilon(3S)\rightarrow \gamma\chi_{b2}(1P))=212$ eV, $\Gamma_0(\Upsilon(3S)\rightarrow \gamma\chi_{b0}(1P))=54$ eV, which are in good agreement with the CLEO and BaBar data, and also to $\Gamma_1(\Upsilon(3S)\rightarrow \gamma\chi_{b1}(1P))=13$ eV, which satisfies the BaBar limit, $\Gamma_1(exp.) < 22$ eV.