CP violation effects on the measurement of $\gamma$ from $B\to DK$

Inspired by the unexpectedly large difference between the CP violation of $D$ decays into $K^{+}{K}^{-}$ and ${\pi }^{+}{\pi }^{-}$, we explore the impact on the extraction of $\gamma$ via the $B\to DK$ process with the $D$ meson reconstructed in the $K^{+}{K}^{-},{\pi }^{+}{\pi }^{-}$ final state. We show that the extracted results for $\gamma$ can be shifted by $\mathcal{O}(A_{CP}\mathrm{/}{r}_B^K)$, where $A_{CP}$ is the direct CP asymmetry in $D$ decays and $r_B^K$ is the ratio of the decay amplitudes of $B^{-}\to {\bar{D}}^0{K}^{-}$ and $B^{-}\to D^0{K}^{-}$. Using the recent data on CP asymmetry, we demonstrate the correction to physical observables in $B\to DK$canreach6$\gamma$byroughly$5^\circ$\mathrm{.}Theremanentcorrectionsdependonthestrongphaseofthe$D$decays,butarelessthan$0.5^\circ$\mathrm{.}Withtheincreasingprecisioninthe$\gamma$ determination on the LHCb experiment and Super B factories, the inclusion of CP violation of $D$ decays will therefore soon become important.