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

Abstract

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 $\stackrel{(-)}{B}$$^{0}_{d}$ mesons. We point out that a clean signal of $CP$ violation with magnitude of 2Re($ε$) $\approx 3.3\times 10^{-3}$ can manifest in the semileptonic decays $D^{+}\rightarrow l^{+}ν^{~}_{l}K_{S}$ ($K_{L}$) vs $D^{-}\rightarrow l^{-}\barν^{~}_{l}K_{S}$ ($K_{L}$). The $CP$ asymmetries are also dominated by 2Re($ε$) in the two-body nonleptonic decays $D^{\pm} \rightarrow (π^{\pm}, ρ^{\pm}, a^{\pm}_{1})+ (K_{S}, K_{L})$ and in the multi-body processes $D^{\pm}\rightarrow π^{\pm}+n_{0}π^{0}+n_{\pm} (π^{\pm}π^{\mp})+ (K_{S}, K_{L})$, 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 $τ$-charm factories or hadron machines. Finally we show that the $CP$ asymmetry induced by Re($ε$) may compete with those from $B^{0}_{d}-\bar{B}^{0}_{d}$ mixing and final-state interactions in the semi-inclusive and exclusive decays $B_{d}\rightarrow X(c\bar{c})+(K_{S}, K_{L})$ on the $Υ(4S)$ resonance.