The NLO QCD corrections to $B_c$($B_c^*$) production around $Z$ pole at an $e^+e^-$ collider

Production of the $B_c$ and $B_c^*$ mesons at $Z$-factory (an $e^+e^-$ collider running at energies around $Z$ pole) is calculated up-to the next-to-leading order (NLO) QCD corrections. The results show that the dependence of the total cross sections on the renormalization scale $\mu$ is reduced by the corrections, and the NLO corrections enhance the production cross section by $52\%$($33\%$) when the renormalization scale is taken at $\mu=2m_b$. For experimental observations of the $B_c$ and $B_c^*$ mesons, the differential cross sections for the production are also analyzed up-to the NLO accuracy.