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Decays of the Meson $B_c$ to a $P$-Wave Charmonium State $χ_c$ or $h_c$

The semileptonic decays, $B_{c}{\longrightarrow}{χ_c}(h_c)+{\ell}+{ν}_{\ell}$, and the two-body nonleptonic decays, $B_{c}{\longrightarrow}{χ_c}(h_c)+h$, (here $χ_c$ and $h_c$ denote $(c\bar c[^3P_J])$ and $(c\bar c[^1P_1])$ respectively, and $h$ indicates a meson) were computed. All of the form factors appearing in the relevant weak-current matrix elements with $B_c$ as its initial state and a $P$-wave charmonium state as its final state for the decays were precisely formulated in terms of two independent overlapping-integrations of the wave-functions of $B_c$ and the $P$-wave charmonium and with proper kinematics factors being `accompanied'. We found that the decays are quite sizable, so they may be accessible in Run-II at Tevatron and in the foreseen future at LHC, particularly, when BTeV and LHCB, the special detectors for B-physics, are borne in mind. In addition, we also pointed out that the decays $B_c\to h_c+...$ may potentially be used as a fresh window to look for the $h_c$ charmonium state, and the cascade decays, $B_c\to χ_c[^3P_{1,2}]+l+ν_l$ ($B_c\to χ_c[^3P_{1,2}]+h$) with one of the radiative decays $χ_c[^3P_{1,2}] \to J/ψ+γ$ being followed accordingly, may affect the observations of $B_c$ meson through the decays $B_{c}\to {J/ψ}+{l}+ν_{l}$ ($B_c\to J/ψ+h$) substantially.