Exploring the $\Upsilon(4S,5S,6S) \to h_b(1P)\eta$ hidden-bottom hadronic transitions

Recently, Belle Collaboration has reported the measurement of the spin-flipping transition $\Upsilon(4S) \to h_b(1P)\eta$ with an unexpectedly large branching ratio: $\mathcal{B}(\Upsilon(4S) \to h_b(1P)\eta) =(2.18\pm 0.11\pm 0.18)\times 10^{-3}$. Such a large branching fraction contradicts with the anticipated suppression for the spin flip. In this work, we examine the effects induced by intermediate bottomed meson loops and point out that these effects are significantly important. Using the effective Lagrangian approach (ELA), we find the experimental data on $\Upsilon(4S) \to h_b(1P)\eta$ can be accommodated with the reasonable inputs. We then explore the decays $\Upsilon(5S,6S)\to h_b(1P)\eta$ and find that these two channels also have sizable branching fractions. We also calculate these these processes in the framework of nonrelativistic effective field theory (NREFT). For the decays $\Upsilon(4S) \to h_b(1P) \eta$, the NREFT results are at the same order of magnitude but smaller than the ELA results by a factor of $2$ to $5$. For the decays $\Upsilon(5S, 6S) \to h_b(1P) \eta$ the NREFT results are smaller than the ELA results by approximately one order of magnitude. We suggest future experiment Belle-II to search for the $\Upsilon(5S, 6S)\to h_b(1P) \eta$ decays which will be helpful to understand the transition mechanism.