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Production of the $η_b(nS)$ states

The rates for magnetic dipole (M1) transitions $Υ(nS) \to η_b(n'S) + γ$, $n' \le n$, 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 $η_b(1S,2S)$. The question then arises whether $Υ(2S)$ or $Υ(3S)$ is a better source of $η_b(1S)$. Whereas one nonrelativistic model favors $η_b(1S)$ production from $Υ(2S)$, 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 $η_b(1S)$ in $Υ(3S)$ radiative decays could be comparable to those in $Υ(2S)$ decays. We also discuss a suggestion for discovering the $η_b$ via $Υ(3S) \to h_b(^1P_1) ππ$, followed by $h_b \to η_b γ$.