Production of the $P$-Wave Excited $B_c$-States through the $Z^0$ Boson Decays

In Ref.[7],we have dealt with the production of the two color-singlet $S$-wave $(c\bar{b})$-quarkonium states $B_c(\mathrm{\mid }(c\bar{b}{)}_{\mathbf{1}}{[}^1{S}_0]>)$ and $B_c^{\ast }(\mathrm{\mid }(c\bar{b}{)}_{\mathbf{1}}{[}^3{S}_1]>)$ through the $Z^0$ boson decays. As an important sequential work, we make a further discussion on the production of the more complicated $P$-wave excited $(c\bar{b})$-quarkonium states, i.e. $\mathrm{\mid }(c\bar{b}{)}_{\mathbf{1}}{[}^1{P}_1]>$ and $\mathrm{\mid }(c\bar{b}{)}_{\mathbf{1}}{[}^3{P}_J]>$ (with $J=(1,2,3)$). More over, we also calculate the channel with the two color-octet quarkonium states $\mathrm{\mid }(c\bar{b}{)}_{\mathbf{8}}{[}^1{S}_0]g>$ and $|(c\bar{b})_{\bf 8}[^3S_1]g>$, whose contributions to the decay width maybe at the same order of magnitude as that of the color-singlet $P$-wave states according to the naive nonrelativistic quantum chromodynamics scaling rules. The $P$-wave states shall provide sizable contributions to the $B_c$ production, whose decay width is about 20% of the total decay width ${\mathrm{\Gamma }}_{Z^0\to B_c}$. After summing up all the mentioned $(c\bar{b})$-quarkonium states’ contributions, we obtain ${\mathrm{\Gamma }}_{Z^0\to B_c}=235.9_{-122.0}^{+352.8}$ KeV, where the errors are caused by the main uncertainty sources.