A difference of several tenths of a percent has been observed between the direct $CP$ asymmetries of $D^0\to K^{+}{K}^{-}$ and $D^0\to {\pi }^{+}{\pi }^{-}$. It has been noted recently that $CP$ asymmetries in such singly Cabibbo-suppressed decays can affect the determination of the weak phase $\gamma$ using the Gronau-London-Wyler method of comparing rates for $B^{+}\to D{K}^{+}$ and $B^{-}\to D{K}^{-}$, where $D$ is a superposition of $D^0$ and ${\overline{D}}^0$ decaying to a $CP$ eigenstate. Using an analysis of the $CP$ asymmetries in singly Cabibbo-suppressed decays based on a $c\to u$ penguin amplitude with a standard model weak phase but enhanced by $CP$-conserving strong interactions, we estimate typical shifts in $\gamma$ of several degrees and pinpoint measurements which would reduce uncertainties to an acceptable level.

• Received 28 January 2013

DOI:https://doi.org/10.1103/PhysRevD.87.074002