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In this paper, we calculate the decay widths and branching fractions for thedecays $Z \to H+ \gamma$ up to ${\cal O}(\alpha_{s} v^2)$ accuracy within theframework of nonrelativistic QCD, where $H$ stands for the $S$-wave quarkonium$\eta_c$, $J/\psi$, $\eta_b$ or $\Upsilon$, respectively. To compare with theleading-order terms, those corrections show good perturbative behavior asexpected. It is found that contributions from the next-to-leading order QCDcorrection ${\cal O}(\alpha_{s}v^0)$, the relativistic correction ${\calO}(\alpha^{0}_{s}v^2)$ 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. Fora high luminosity electron-positron collider running around the $Z$-pole, dueto $Z$-boson resonance effect, sizable events could be produced from those raredecay channels.

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