The $B_c\rightarrow ψ(2S)π$, $η_c(2S)π$ decays in the perturbative QCD approach

Nonleptonic two body $B_c$ decays including radially excited $ψ(2S)$ or $η_c(2S)$ mesons in the final state are studied using the perturbative QCD approach based on $k_T$ factorization. The charmonium distribution amplitudes are extracted from the $n = 2, l = 0$ Schr$\ddot{o}$dinger states for the harmonic oscillator potential. Utilizing these distribution amplitudes, we calculate the numerical results of the $B_c\rightarrow ψ(2S),η_c(2S)$ transition form factors and branching fractions of $B_c\rightarrow ψ(2S)π, η_c(2S)π$ decays. The ratio between two decay modes $B_c\rightarrow ψ(2S)π$ and $B_c\rightarrow J/ψπ$ is compatible with the experimental data within uncertainties, which indicate that the harmonic oscillator wave functions for $ψ(2S)$ and $η_c(2S)$ work well. It is found that the branching fraction of $B_c\rightarrow η_c(2S)π$, which is dominated by the twist-3 charmonium distribution amplitude, can reach the order of $10^{-3}$. We hope it can be measured soon in the LHCb experiment.

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