Production of the $\eta_b(nS)$ states

The rates for magnetic dipole (M1) transitions $\Upsilon(nS) \to \eta_b(n'S) + \gamma$, $n' \le n$, are compared. The photon energies for allowed ($n' = n$) M1 transitions are very small, so hindered ($n' < n$) transitions could be more favorable for discovering the $\eta_b(1S,2S)$. The question then arises whether $\Upsilon(2S)$ or $\Upsilon(3S)$ is a better source of $\eta_b(1S)$. Whereas one nonrelativistic model favors $\eta_b(1S)$ production from $\Upsilon(2S)$, this advantage is lost when relativistic corrections are taken into account, and is not common to all sets of wave functions even in the purely nonrelativistic limit. Thus, the prospects for discovering $\eta_b(1S)$ in $\Upsilon(3S)$ radiative decays could be comparable to those in $\Upsilon(2S)$ decays. We also discuss a suggestion for discovering the $\eta_b$ via $\Upsilon(3S) \to h_b(^1P_1) \pi \pi$, followed by $h_b \to \eta_b \gamma$.