$Z_0$ Boson Decays to $B_c^{(\ast )}$ Meson and Its Uncertainties

The programming new $e^{+}{e}^{-}$ collider with high luminosity shall provide another useful platform to study the properties of the doubly heavy $B_c$ meson in addition to the hadronic colliders as LHC and TEVATRON. Under the `New Trace Amplitude Approach’, we calculate the production of the spin-singlet $B_c$ and the spin-triplet $B_c^{\ast }$ mesons through the $Z^0$ boson decays, where uncertainties for the production are also discussed. Our results show ${\mathrm{\Gamma }}_{{(}^1{S}_0)}=81.4_{-40.5}^{+102.1}$ KeV and ${\mathrm{\Gamma }}_{{(}^3{S}_1)}=116.4_{-62.8}^{+163.9}$ KeV, where the errors are caused by varying $m_b$ and $m_c$ within their reasonable regions.