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It has been found that at a high luminosity $e^+ e^-$ collider, sizable$\eta_c+c\bar{c}X$ and $\eta_b+b\bar{b}X$ events can be produced when it worksaround the $Z$ peak. In this paper, we calculate the decay widths of $Z \to\eta_c+c+\bar{c}+X$ and $Z \to \eta_b+b+\bar{b}+X$ up to next-to-leading order(NLO) accuracy. We find that the NLO corrections are significant in these twoprocesses. After including the NLO corrections, the decay widths of $Z \to\eta_c+c+\bar{c}+X$ and $Z \to \eta_b+b+\bar{b}+X$ are enhanced by about $37\%$and $28\%$ for the case of $\mu_R=2m_c$ and $\mu_R=2m_b$, respectively. Thedifferential decay widths $d\Gamma/ds_1$ and $d\Gamma/dz$ for these two decayprocesses are also analyzed.

Next-to-leading order QCD corrections to $Z\to \eta_Q+Q+\bar{Q}$