$B^{\pm }\to D{K}^{\pm }$ with direct $CP$ violation in charm

We investigate the implications of direct $CP$ violation (CPV) on the determination of the unitarity triangle angle $\gamma$ from $B\to DK$ decays. We show that $\gamma$ can still be extracted even with the inclusion of direct CPV in charm if (i) at least one of the $D$ decays has negligible $CP$ violation; and (ii) data from a charm factory at threshold are used. If approximate expressions without direct $CP$ violation in charm are used, this can result in a shift in $\gamma$ that is $\mathcal{O}(r_D/r_B)$. It is modest for $B\to DK$ but can be $\mathcal{O}(1)$ for $B\to D\pi$. We illustrate the size of the shift using an example of the Gronau-London-Wyler method.

Figure 1

The orientation of the triangles in the GLW method with $CP$ violation in SCS $D$ decays included.