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Nonleptonic two body $B_c$ decays including radially excited $\psi(2S)$ or$\eta_c(2S)$ mesons in the final state are studied using the perturbative QCDapproach based on $k_T$ factorization. The charmonium distribution amplitudesare extracted from the $n = 2, l = 0$ Schr$\ddot{o}$dinger states for theharmonic oscillator potential. Utilizing these distribution amplitudes, wecalculate the numerical results of the $B_c\rightarrow \psi(2S),\eta_c(2S)$transition form factors and branching fractions of $B_c\rightarrow \psi(2S)\pi,\eta_c(2S)\pi$ decays. The ratio between two decay modes $B_c\rightarrow\psi(2S)\pi$ and $B_c\rightarrow J/\psi\pi$ is compatible with the experimentaldata within uncertainties, which indicate that the harmonic oscillator wavefunctions for $\psi(2S)$ and $\eta_c(2S)$ work well. It is found that thebranching fraction of $B_c\rightarrow \eta_c(2S)\pi$, which is dominated by thetwist-3 charmonium distribution amplitude, can reach the order of $10^{-3}$. Wehope it can be measured soon in the LHCb experiment.

The $B_c\rightarrow \psi(2S)\pi$, $\eta_c(2S)\pi$ decays in the perturbative QCD approach