  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 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 $B_c\rightarrow \psi(2S),\eta_c(2S)$
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 experimental
data within uncertainties, which indicate that the harmonic oscillator wave
functions for $\psi(2S)$ and $\eta_c(2S)$ work well. It is found that the
branching fraction of $B_c\rightarrow \eta_c(2S)\pi$, 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.

High Energy Physics - Phenomenology