$Z$-boson decays into $S$-wave quarkonium plus a photon up to ${\cal O}(\alpha_{s} v^2)$ corrections

In this paper, we calculate the decay widths and branching fractions for the decays $Z \to H+ \gamma$ up to ${\cal O}(\alpha_{s} v^2)$ accuracy within the framework of nonrelativistic QCD, where $H$ stands for the $S$-wave quarkonium $\eta_c$, $J/\psi$, $\eta_b$ or $\Upsilon$, respectively. To compare with the leading-order terms, those corrections show good perturbative behavior as expected. It is found that contributions from the next-to-leading order QCD correction ${\cal O}(\alpha_{s}v^0)$, the relativistic correction ${\cal O}(\alpha^{0}_{s}v^2)$ and their joint correction ${\cal O}(\alpha_{s} v^2)$ are sizable and comparable to each other, especially for the charmonium case. Thus we need to take all of them into consideration for a sound estimation. For a high luminosity electron-positron collider running around the $Z$-pole, due to $Z$-boson resonance effect, sizable events could be produced from those rare decay channels.