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Within the framework of nonrelativistic QCD (NRQCD) factorization formalism,we compute QCD next-to-next-to-leading order (NNLO) corrections to the helicityamplitudes as well as the decay width of $Z\to H+\gamma$, where $H$ can be$\eta_Q (Q=c,b), J/\psi$, or $\Upsilon$. In addition, we resum thenext-to-leading logarithms (NLL) of ${m_Z^2}/{m_Q^2}$ to all orders of$\alpha_s$ for the leading-twist helicity amplitude by employing the light-conefactorization approach. It is worth mentioning that we obtain the analyticexpressions of the truncated NLL at $\alpha_s^2$. We find that the$\mathcal{O}(\alpha_s)$ corrections are around 10\% for $\eta_c$ and $\Upsilon$productions, however insignificant for $J/\psi$ and $\eta_b$ productions. The$\mathcal{O}(\alpha_s^2)$ corrections are moderate for charmonium production,while very small for bottomonium production. Moreover, it is found that the NLLresummation can considerably alter the NRQCD prediction, especially for$J/\psi$ production. Combining the NRQCD and light-cone computation, we makephenomenological predictions on the decay widths and branching fractions. Inaddition, we investigate the dependence of the theoretical results on the heavyquark mass, and find the branching fraction of $Z\to H+\gamma$ monotonicallydecreases as $m_Q$ increases.

$Z$ boson radiative decays to a $S$-wave quarkonium at NNLO and NLL accuracy