Prediction of $B_c\to D\pi$ in the pQCD approach

We investigate the branching ratios and direct CP asymmetries of $B_c^+\to D^0\pi^+$ and $B_c^+\to D^+\pi^0$ decays in the PQCD approach. All the diagrams with emission topology or annihilation topology are calculated strictly. A branching ratio of $10^{-6}$ and $10^{-7}$ for $B_c^+\to D^0\pi^+$ and $B_c^+\to D^+\pi^0$ decay is predicted, respectively. Because of the different weak phase and strong phase from penguin operator and two kinds of tree operator contributions, we predict a possible large direct CP violation: $A^{\rm dir}_{cp} (B_c^\pm \to D^0 \pi^\pm)\approx -50$ and $A^{\rm dir}_{cp} (B_c^\pm \to D^\pm \pi^0)\approx 25$ when $\gamma=55^\circ$, which can be tested in the coming LHC.