Study of $B^\pm_c \to (D^0 K^\pm, D^0 \pi^\pm)$ decays

LHCb observes the $B^+_c \to D^0 K^+$ decay with $R_{D^0 K} = f_c/f_u \times {\cal B}(B^+_c \to D^0 K^+)=(9.3^{+2.8}_{-2.5} \pm 0.6)\times 10^{-7}$. The corresponding branching ratio (BR) of the decay can be estimated as ${\cal B}(B^+_c \to D^0 K^+) \approx (10.01 \pm 3.40)\times 10^{-5}$; however, the theoretical estimates vary from $\sim 10^{-7}$ to $\sim 5\times 10^{-5}$. We phenomenologically investigate the $B^+_c \to (D^0 K^+, D^0 \pi^+)$ decays through the analysis of $B\to KK$, $B^+_u\to D^+ K^0$, and $B_d \to D^-_s K^+$. With the form factor of $f^{B_c D}_0\approx 0.2$, it is found that the tree-annihilation contribution dominates the $B^+_c \to D^0 K^+$ decay, and when ${\cal B}(B^+_u \to D^+ K^0) \approx (1-3.1) \times 10^{-7}$ is required, we obtain ${\cal B}(B^+_c \to D^0 K^+)\approx (4.4- 9) \times 10^{-5}$, and the magnitude of CP asymmetry is lower than approximately $10\%$. Although the $B^+_c \to D^0 \pi^+$ decay is dominated by the tree-transition effect, the tree-annihilation also makes an important contribution, where its effect could be around $70\%$ of the tree-transition. It is found that when ${\cal B}(B^+_c \to D^0 K^+)\approx (4.4- 9) \times 10^{-5}$ is taken, the BR and CP asymmetry for $B^+_c \to D^0 \pi^+$ with the common values of parameters can be ${\cal B}(B^+_c \to D^0 \pi^+)\approx (4.9-8)\times 10^{-6}$ and of the order of one, respectively. Moreover, we conclude ${\cal B}(B^+_c \to D^+ K^0)\approx {\cal B}(B^+_c \to D^0 K^+)$, and the BRs for $B^+_c \to K^+ \bar K^0$ and $B^+_c \to J/\Psi \pi^+$ are $(6.99 \pm 1.34) \times 10^{-7}$ and $(7.7 \pm 1.1)\times 10^{-4}$, respectively.