Possible Effects of $D^0-\bar{D}^0$ Mixing in Weak $B$ Decays

$D^0-\bar{D}^0$ mixing at a detectable level requires the presence of new physics and may lead to some observable effects in weak decays of $B$ mesons. We show that $CP$ violation induced by $D^0-\bar{D}^0$ mixing can manifest itself in the decay-rate asymmetry of $B^+_u\rightarrow D_{L(H)}l^+\nu^{~}_l$ vs $B^-_u\rightarrow D_{L(H)}l^-\bar{\nu}^{~}_l$. A rephasing-invariant generalization of the Gronau-Wyler approach is made to determine the weak phase shift in $B^{\pm}_u\rightarrow D_{L(H)}K^{\pm}$, which is only sensitive to the underlying new physics in $D^0-\bar{D}^0$ mixing. We also demonstrate the possible effect of $D^0-\bar{D}^0$ mixing on $CP$ violation in decay modes of the type $B_d\rightarrow D_{L(H)}+(\pi^0, \rho^0, {\rm etc})$. Finally the model of four quark families is taken as an example to illustrate how the new physics affects $D^0-\bar{D}^0$ mixing, $B^0_d-\bar{B}^0_d$ mixing and $CP$ asymmetries in the relevant $B$ decays.