Prediction of anomalous $\Upsilon(5S)\to\Upsilon(1^3D_J)\eta$ transitions

In this work, we study the hadronic loop contribution to the $\Upsilon(5S)\to \Upsilon(1^3D_J)\eta$ ($J=1,2,3$) transitions. We predict that the branching ratios of $\Upsilon(5S)\to \Upsilon(1^3D_1)\eta$, $\Upsilon(5S)\to \Upsilon(1^3D_2)\eta$ and $\Upsilon(5S)\to \Upsilon(1^3D_3)\eta$ can reach up to $(0.5\sim5.1)\times10^{-3}$, $(0.7\sim7.5)\times10^{-3}$ and $(0.9\sim9.6)\times10^{-4}$, respectively. Since these predicted hadronic transitions of $\Upsilon(5S)$ are comparable with these observed $\Upsilon(5S)\to \Upsilon(nS)\pi^+\pi^-$ $(n=1,2,3)$, we suggest future experiment like Belle and BelleII to carry out the search for these anomalous $\Upsilon(5S)\to \Upsilon(1^3D_J)\eta$ transitions.