Estimating of CP Violation in $Bc→ B K⁰+B {K̄}^0→ B π± e∓ νₑ$ Decays

In this paper, we investigate the CP asymmetries ACPᵖᵐ and ACPᵐᵖ in Bc±→ B± K⁰+B± K̄⁰→ B± π± e∓ νₑ and Bc±→ B± K⁰+B± K̄⁰→ B± π∓ e± νₑ decays, both of them consist of three parts: the indirect CP violations in K⁰ -K̄⁰ mixing ACP,mixᵖᵐ and ACP,mixᵐᵖ, the direct CP violations in Bc decay ACP,dirᵖᵐ and ACP,dirᵐᵖ, the new CP violation effects ACP,intᵖᵐ and ACP,intᵐᵖ, which originate from the interference between the amplitude of the Bc⁻→ B⁻ K⁰ (K̄⁰) → B⁻ π⁻ e⁺ νₑ(π⁺ e⁻ ν̄ₑ) decay with the difference between the oscillating effect of K⁰→ K̄⁰ and that of K̄⁰→ K⁰. The strong phase differences of the direct CP asymmetries ACP,dirᵖᵐ and ACP,dirᵐᵖ arising from K⁰-K̄⁰ mixing parameters. ACPᵖᵐ and ACPᵐᵖ are of the order O(10⁻⁴) and O(10⁻³), respectively. The new CP violation effect ACP,intᵖᵐ plays a dominant role in ACPᵖᵐ, the CP asymmetry ACPᵐᵖ is dominated by the indirect CP violation ACP,mixᵐᵖ, so the CP asymmetry ACPᵖᵐ provides an ideal place to study the new CP violation effect. We derive another two asymmetry observables ACPᵖᵖ and ACPᵐᵐ, which are dominated by K⁰-K̄⁰ mixing. The observables ACPᵖᵐ, ACPᵐᵖ, ACPᵖᵖ and ACPᵐᵐ are hopefully to be marginally observed at the LHC experiment and the HL-LHC experiment.