Exclusive production of $B_c$ mesons in $e^{+}{e}^{-}$ colliders

Within the framework of the perturbative QCD approach, we calculate the time-like $B_c{B}_c(B_c^{\ast })$ form factors $F(Q^2)$ and $A_2(Q^2)$. We include relativistic corrections and QCD corrections, either of which can give about $20\mathrm{\%}$ correction to the leading-order contribution, but there are cancellation effects between them. We calculate the cross sections of the $e^+e^-\to B_c^-B_c^+(B_c^{*+})$processes\mathrm{.}Thecrosssectionsareenhancedatthe$Z$poletobe$\sigma^{PP}(Q=m_Z) \sim 1.3\times10^{-5}$\text{pb}$$and$\sigma^{PV}(Q=m_Z) \sim 2.5\times10^{-5}$\text{pb}$$, which are still too small to be detected by proposed $e^{+}{e}^{-}$ colliders such as the Circular Electron Positron Collider.