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

In this paper, we calculate the decay widths and branching fractions for the decays $Z\to H+\gamma$ up to $\mathcal{O}({\alpha }_s{v}^2)$ accuracy within the framework of nonrelativistic QCD, where $H$ stands for the $S$-wave quarkonium ${\eta }_c$, $J\mathrm{/}\psi$, ${\eta }_b$ or $\mathrm{{\rm Y}}$, 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 $\mathcal{O}({\alpha }_s{v}^0)$, the relativistic correction $\mathcal{O}({\alpha }_s^0{v}^2)$ and their joint correction $\mathcal{O}({\alpha }_s{v}^2)$ 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.