$B_c$ Meson Production around the $Z^0$ Peak at a High Luminosity $e^+ e^-$ Collider

Considering the possibility to build an $e^+ e^-$ collider at the energies around the $Z^0$-boson resonance with a planned luminosity so high as ${\cal L}\propto 10^{34}\sim 10^{36}cm^{-2}s^{-1}$ (super $Z$-factory), we make a detailed discussion on the $(c\bar{b})$-quarkonium production through $e^{+}+e^{-}\rightarrow (c\bar{b})[n]+b+\bar{c}$ within the framework of non-relativistic QCD. To simplify the hard-scattering amplitude as much as possible and to derive analytic expressions for the purpose of future events simulation, we adopt the "improved trace technology" to do our calculation, which deals with the hard scattering amplitude directly at the amplitude level other than the conventional way at the squared-amplitude level. Total cross-section uncertainties caused by the quark masses are predicted by taking $m_c=1.50\pm0.30$ GeV and $m_b=4.90\pm0.40$ GeV. If all higher $(c\bar{b})$-quarkonium states decay to the ground state $B_c$ ($|(c\bar{b})_{\bf 1}[^1S_0]>$) with 100% efficiency, we obtain $\sigma_{e^{+}+e^{-}\rightarrow B_{c}+b+\bar{c}} =5.190^{+6.222}_{-2.419}$ pb, which shows that about $10^5 \sim 10^7$ $B_c$ events per operation year can be accumulated in the super $Z$-factory. If taking the collider energy runs slightly off the $Z^0$-peak, i.e. $\sqrt{S}=(1.00\pm0.05) m_Z$, the total cross-section shall be lowered by about one-order from its peak value. Such a super $Z$-factory shall provide another useful platform to study the properties of $B_c$ meson, or even the properties of its excited $P$-wave states, in addition to its production at the hadronic colliders Tevatron and LHC.