Study of CP Violation in $D^{\pm}\rightarrow K^{\ast}(892)^{0} \pi^{\pm} + \bar{K}^{\ast }(892)^{0}\pi^{\pm}\rightarrow K_{S,L}^{0}\pi^0 \pi^{\pm}$ Decays

Within the Standard Model, we investigate the CP violations and the $K_S^0-K_L^0$ asymmetries in $D^{\pm}\rightarrow K^{\ast}(892)^{0} \pi^{\pm} + \bar{K}^{\ast }(892)^{0}\pi^{\pm}\rightarrow K_{S,L}^{0}\pi^0 \pi^{\pm}$ decays basing on the factorization-assisted topological-amplitude (FAT) approach and the topological amplitude (TA) approach of Cheng and Chiang [Phys. Rev. D 104, 073003 (2021).]. We find that the CP violations in these decays $A_{CP}^{K_{S,L}^0}$ can exceed the order of $10^{-3}$ in the two approaches and consist of three parts: the indirect CP violations in $K^0 -\bar{K}^0$ mixing $A_{CP,K_{S,L}^0}^{mix}$, the direct CP violations in charm decays $A_{CP,K_{S,L}^0}^{dir}$ and the new CP violation effects $A_{CP,K_{S,L}^0}^{int}$, which are induced from the interference between two tree (Cabibbo-favored and doubly Cabibbo-suppressed) amplitudes with the neutral kaon mixing. The indirect CP violations in $K^0 -\bar{K}^0$ mixing play a dominant role in $A_{CP}^{K_{S,L}^0}$, the new CP violation effects have a non-negligible contribution to $A_{CP}^{K_{S,L}^0}$. We estimate the numerical results of the $K_S^0-K_L^0$ asymmetries $R_{K_{S}-K_{L}}^{D^{\pm}}$ and find that there exist a large difference between the numerical results in the FAT approach and that of the TA approach. We also find that if ones adopt the values of the decay time parameters $t_0 = 3.0 \tau_S$ and $t_1 = 10.0 \tau_S$, the new CP violation effect $A_{CP,K_{S}^0}^{int}$ would dominate the CP violation in $D^{\pm} \rightarrow K^{\ast}(892)^{0} \pi^{\pm} + \bar{K}^{\ast }(892)^{0}\pi^{\pm}\rightarrow K_{S}^0\pi^0 \pi^{\pm}$ decays and could be observed with $6.7\times 10^{6}$ and $6.5\times 10^{6}$ $D^{\pm}$ events-times-efficiency in the FAT approach and the TA approach, respectively. Our results could be tested by the LHCb, Belle II and BESIII experiments.