Study of $B_{c} \rightarrow \psi(2S) K$, $\eta_{c}(2S)K$, $\psi(3770)K$ decays with perturbative QCD approach

We study the $B_{c}$$\rightarrow$$\psi(2S)$K, $\eta_{c}(2S)$K, $\psi(3770)$K decays with perturbative QCD approach (pQCD) based on $k_T$ factorization. The new orbitally excited charmonium distribution amplitudes $\psi(1^{3}D_{1})$ based on the Schr\"{o}dinger wave function of the $n=1$, $l=2$ state for the harmonic-oscillator potential are employed. By using the corresponding distribution amplitudes, we calculate the branching ratio of $B_{c}$$\rightarrow$$\psi(2S)$K, $\eta_{c}(2S)$K, $\psi(3770)$K decays and the form factors $A_{0,1,2}$ and $V$ for the transition $B_{c}$$\rightarrow$$\psi(1^{3}D_{1})$. We obtain the branching ratio of both $B_{c}$$\rightarrow$$\psi(2S)$K and $B_{c}$$\rightarrow$$\eta_{c}(2S)$K are at the order of $10^{-5}$. The effects of two sets of the S-D mixing angle $\theta=-12^{\circ}$ and $\theta=27^{\circ}$ for the decay $B_{c}$$\rightarrow$$\psi(3770)$K are studied firstly in this paper. Our calculations show that the branching ratio of the decay $B_{c}$$\rightarrow$$\psi(3770)$K can be raised from the order of $10^{-6}$ to the order of $10^{-5}$ at the mixing angle $\theta=-12^{\circ}$, which can be tested by the running LHC-b experiments.