Electroweak effects in the extraction of the CKM angle $\gamma$ from $B\to D\pi$ decays

The angle $\gamma$ of the standard CKM unitarity triangle can be determined from tree-level $B$-meson decays essentially without hadronic uncertainties. We calculate the second-order electroweak corrections for the $B\to D\pi$ modes and show that their impact on the determination of $\gamma$ could be enhanced by an accidental cancellation of poorly known hadronic matrix elements. However, we do not expect the resulting shift in $\gamma$ to exceed $\big|\delta \gamma^{D\pi} /\gamma \big| \lesssim {\mathcal O}\big(10^{-4}\big)$.