Estimating of CP Violation in $B_c\to B{K}^0+B{\bar{K}}^0\to B{\pi }^{\pm }{e}^{\mp }{\nu }_e$ Decays

Abstract

In this paper, we investigate the CP asymmetries ${\mathcal{A}}_{CP}^{pm}$ and ${\mathcal{A}}_{CP}^{mp}$ in $B_c^{\pm }\to B^{\pm }{K}^0+B^{\pm }{\bar{K}}^0\to B^{\pm }{\pi }^{\pm }{e}^{\mp }{\nu }_e$ and $B_c^{\pm }\to B^{\pm }{K}^0+B^{\pm }{\bar{K}}^0\to B^{\pm }{\pi }^{\mp }{e}^{\pm }{\nu }_e$ decays, both of them consist of three parts: the indirect CP violations in $K^0-{\bar{K}}^0$ mixing $A_{CP,mix}^{pm}$ and $A_{CP,mix}^{mp}$, the direct CP violations in $B_c$ decay $A_{CP,dir}^{pm}$ and $A_{CP,dir}^{mp}$, the new CP violation effects $A_{CP,int}^{pm}$ and $A_{CP,int}^{mp}$, which originate from the interference between the amplitude of the $B_c^{-}\to B^{-}{K}^0({\bar{K}}^0)\to B^{-}{\pi }^{-}{e}^{+}{\nu }_e({\pi }^{+}{e}^{-}{\bar{\nu }}_e)$ decay with the difference between the oscillating effect of $K^0\to {\bar{K}}^0$ and that of ${\bar{K}}^0\to K^0$. The strong phase differences of the direct CP asymmetries $A_{CP,dir}^{pm}$ and $A_{CP,dir}^{mp}$ arising from $K^0-{\bar{K}}^0$ mixing parameters. ${\mathcal{A}}_{CP}^{pm}$ and ${\mathcal{A}}_{CP}^{mp}$ are of the order $\mathcal{O}(10^{-4})$ and $\mathcal{O}(10^{-3})$, respectively. The new CP violation effect ${\mathcal{A}}_{CP,int}^{pm}$ plays a dominant role in ${\mathcal{A}}_{CP}^{pm}$, the CP asymmetry ${\mathcal{A}}_{CP}^{mp}$ is dominated by the indirect CP violation ${\mathcal{A}}_{CP,mix}^{mp}$, so the CP asymmetry ${\mathcal{A}}_{CP}^{pm}$ provides an ideal place to study the new CP violation effect. We derive another two asymmetry observables ${\mathcal{A}}_{CP}^{pp}$ and ${\mathcal{A}}_{CP}^{mm}$, which are dominated by $K^0-{\bar{K}}^0$ mixing. The observables ${\mathcal{A}}_{CP}^{pm}$, ${\mathcal{A}}_{CP}^{mp}$, ${\mathcal{A}}_{CP}^{pp}$ and ${\mathcal{A}}_{CP}^{mm}$ are hopefully to be marginally observed at the LHC experiment and the HL-LHC experiment.