Measuring the CKM angle $\gamma$ at LHCb using untagged $B_s \to D \phi$ decays

The potential of a method for the measurement of $\gamma$ using untagged $B_s \to D \phi$ decays is evaluated for the LHCb experiment. In this initial study, the $D$ meson is reconstructed in well-understood decay modes such as $K\pi$, $K\pi\pi\pi$, and $KK$. With realistic assumptions on the detector performance, it is found that the accuracy on $\gamma$ strongly depends on the unknown strong-phase difference, $\delta_B$. A statistical error of $10^{\circ}$ can be achieved in a nominal year of data-taking (2$fb^{-1}$) for favourable values of $\delta_B$. The main systematic uncertainties and the prospects for improving the sensitivity of this method are also briefly discussed.