Study of $B_c\to \psi (2S)K$, ${\eta }_c(2S)K$, $\psi (3770)K$ decays with perturbative QCD approach

We study the $B_c$$\to$$\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$$\to$$\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$$\to$$\psi (1^3{D}_1)$. We obtain the branching ratio of both $B_c$$\to$$\psi (2S)$K and $B_c$$\to$${\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$$\to$$\psi (3770)$K are studied firstly in this paper. Our calculations show that the branching ratio of the decay $B_c$$\to$$\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.