CP violation in quasi-two-body $D\to VP$ decays and three-body $D$ decays mediated by vector resonances

We re-examine direct CP violation in the quasi-two-body $D\to V\!P$ decays and study CP asymmetries in three-body $D$ decays proceeding through intermediate vector resonances within the framework of the topological amplitude approach for tree amplitudes and the QCD factorization approach for penguin amplitudes. Using the same mechanism of incorporating long-distance penguin-exchange amplitude induced from final-state rescattering that nicely accounts for the CP asymmetry difference between $D^0\to K^+K^-$ and $\pi^+\pi^-$ modes, we find that CP asymmetry can also occur at the per mille level in many of the $D\to V\!P$ channels or otherwise be negligibly small. We point out six golden modes which have sufficiently large branching fractions and direct CP violation of order $10^{-3}$. In particular, the direct CP asymmetry difference $\Delta a_{CP}^{V\!P}$ between $D^0\to K^+K^{*-}$ and $\pi^+\rho^-$ is predicted to be $(-1.61\pm0.33)\times 10^{-3}$, very similar to the counterpart in the $P\!P$ sector. Besides, we find a positive and large CP asymmetry of order $10^{-3}$ in $D^0\to K_S K^{*0}$, an example of the asymmetry induced at the tree level. We take into account the flavor-singlet QCD-penguin contributions to calculate their effects on the modes such as $D\to\pi\phi$ and $D\to \eta \phi$. We compare our results with the other approach in detail. For three-body $D$ decays, we perform the Dalitz plot analysis of those decays mediated by several vector resonances with the same mass but different electric charges. Local CP asymmetry varies in magnitude and sign from region to region and can reach the percent level in certain local regions of phase space due to interference.