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

Inspired by the unexpectedly large difference between the CP violation of $D$ decays into $K^+K^-$ and $π^+π^-$, we explore the impact on the extraction of $γ$ via the $B\to DK$ process with the $D$ meson reconstructed in the $K^+K^-,π^+π^-$ final state. We show that the extracted results for $γ$ can be shifted by ${\cal O}(A_{CP} / r_B^K)$, where $A_{CP}$ is the direct CP asymmetry in $D$ decays and $r_B^K$ is the ratio of the decay amplitudes of $B^-\to \bar D^0K^-$ and $B^-\to D^0K^-$. Using the recent data on CP asymmetry, we demonstrate the correction to physical observables in $B\to DK$ can reach 6%, which corresponds to the shift of $γ$ by roughly $5^\circ$. The remanent corrections depend on the strong phase of the $D$ decays, but are less than $0.5^\circ$. With the increasing precision in the $γ$ determination on the LHCb experiment and Super B factories, the inclusion of CP violation of $D$ decays will therefore soon become important.