In this paper, we investigate the $CP$ asymmetries ${\mathcal{A}}_{CP}^{pm}$ and ${\mathcal{A}}_{CP}^{mp}$ in $B_c^{\pm }\to B^{\pm }{K}^0+B^{\pm }{\overline{K}}^0\to B^{\pm }{\pi }^{\pm }{e}^{\mp }{\nu }_e$ and $B_c^{\pm }\to B^{\pm }{K}^0+B^{\pm }{\overline{K}}^0\to B^{\pm }{\pi }^{\mp }{e}^{\pm }{\nu }_e$ decays; both of them consist of three parts: the indirect $CP$ violations in $K^0-{\overline{K}}^0$ mixing $A_{CP,\mathrm{mix}}^{pm}$ and $A_{CP,\mathrm{mix}}^{mp}$, the direct $CP$ violations in $B_c$ decay $A_{CP,\mathrm{dir}}^{pm}$ and $A_{CP,\mathrm{dir}}^{mp}$, and the new $CP$ violation effects $A_{CP,\mathrm{int}}^{pm}$ and $A_{CP,\mathrm{int}}^{mp}$, which originate from the interference between the amplitude of the $B_c^{-}\to B^{-}{K}^0({\overline{K}}^0)\to B^{-}{\pi }^{-}{e}^{+}{\nu }_e({\pi }^{+}{e}^{-}{\overline{\nu }}_e)$ decay with the difference between the oscillating effect of $K^0\to {\overline{K}}^0$ and that of ${\overline{K}}^0\to K^0$. The strong phase differences of the direct $CP$ asymmetries $A_{CP,\mathrm{dir}}^{pm}$ and $A_{CP,\mathrm{dir}}^{mp}$ arise from $K^0-{\overline{K}}^0$ mixing parameters. ${\mathcal{A}}_{CP}^{pm}$ and ${\mathcal{A}}_{CP}^{mp}$ are of the order of $\mathcal{O}({10}^{-4})$ and $\mathcal{O}({10}^{-3})$, respectively. The new $CP$ violation effect ${\mathcal{A}}_{CP,\mathrm{int}}^{pm}$ plays a dominant role in ${\mathcal{A}}_{CP}^{pm}$; the $CP$ asymmetry ${\mathcal{A}}_{CP}^{mp}$ is dominated by the indirect $CP$ violation ${\mathcal{A}}_{CP,\mathrm{mix}}^{mp}$, 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 $K^0-{\overline{K}}^0$ 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.

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• Electroweak interaction
• Electroweak symmetry breaking
• Particle decays
• Particle mixing & oscillations
• Particle phenomena