Study of $B_c^{\pm }\to (D^0{K}^{\pm },D^0{\pi }^{\pm })$ 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 $f_0^{B_{c}D}\approx 0.2$, it is found that the tree-annihilation contribution dominates the $B_c^{+}\to D^0{K}^{+}$ decay, and when $\mathcal{B}(B_u^{+}\to D^{+}{K}^0)\approx (1-3.1)\times {10}^{-7}$ is required, we obtain $\mathcal{B}(B_c^{+}\to D^0{K}^{+})\approx (4.4-9)\times {10}^{-5}$, and the magnitude of CP asymmetry is lower than approximately $10\mathrm{\%}$. 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 $70\mathrm{\%}$ 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 ${\cal B}(B^+_c \to D^+ K^0)\approx {\cal B}(B^+_c \to D^0 K^+)$,andtheBRsfor$B^+_c \to K^+ \bar K^0$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.