Photoproduction of the $B_c$ meson at the future $e^+e^-$ colliders

In the present paper, we study the photoproduction of various $S$-wave $B_c$ states, i.e. $B_c(1{}^1S_0)$, $B_c^*(1{}^1S_0)$, $B_c(2{}^1S_0)$ and $B^*_c(2{}^3S_1)$, at future $e^+e^-$ colliders within the framework of non-relativistic QCD. Two sources of the initial photons can be realized in the production, i.e., the LBS photon and the WWA photon. In addition to the direct photoproduction via the sub-process, $\gamma + \gamma \rightarrow B_c + b + \bar{c}$, we also calculate the resolved photoproduction via the sub-processes, $\gamma + g \rightarrow B_c + b + \bar{c}$ and $q + \bar{q} \rightarrow B_c + b + \bar{c}$ with $q=u,d,s,g$, respectively. Numerical results indicate that the contributions from the single resolved photoproduction are significant and even dominant at certain collision configuration. At the future high-energy and high-luminosity $e^+e^-$ colliders, the $B_c$ meson generated via the photoproduction mechanism is promisingly observable and can be well studied.