Effect of $K^0-{\bar{K}}^0$ Mixing on $CP$ Asymmetries in Weak Decays of $D$ and $B$ Mesons

Within the standard electroweak model we carry out an instructive analysis of the effect of $K^0-{\bar{K}}^0$ mixing on $CP$ asymmetries in some weak decay modes of $D^{\pm }$ and $$${}_d^0$ mesons. We point out that a clean signal of $CP$ violation with magnitude of 2Re($ϵ$) $\approx 3.3\times 10^{-3}$canmanifestinthesemileptonicdecays$D^{+}\rightarrow l^{+}\nu^{~}{l}K{S}$($K_{L}$)vs$D^{-}\rightarrow l^{-}\bar{\nu}^{~}{l}K{S}$($K_{L}$)\mathrm{.}The$CP$ asymmetries are also dominated by 2Re($ϵ$) in the two-body nonleptonic decays $D^{\pm} \rightarrow (\pi^{\pm}, \rho^{\pm}, a^{\pm}{1})+ (K{S}, K_{L})$ and in the multi-body processes $D^{\pm}\rightarrow \pi^{\pm}+n_{0}\pi^{0}+n_{\pm} (\pi^{\pm}\pi^{\mp})+ (K_{S}, K_{L})$,where$n_{0}$and$n_{\pm}$ are integers. It is possible to observe such $CP$-violating signals with about ${10}^7$ $D^{\pm }$ events at $\tau$-charm factories or hadron machines. Finally we show that the $CP$ asymmetry induced by Re($ϵ$) may compete with those from $B_d^0-{\bar{B}}_d^0$ mixing and final-state interactions in the semi-inclusive and exclusive decays $B_{d}\rightarrow X(c\bar{c})+(K_{S}, K_{L})$onthe$\Upsilon (4S)$ resonance.