Measurement of the CKM angle $\gamma$ using $B^\pm \to D K^\pm$ with $D \to K^0_{\rm S} \pi^+\pi^-, K^0_{\rm S} K^+ K^-$ decays

A binned Dalitz plot analysis of $B^\pm \to D K^\pm$ decays, with $D \to K^0_{\rm S} \pi^+\pi^-$ and $D \to K^0_{\rm S} K^+ K^-$, is performed to measure the $C\!P$-violating observables $x_{\pm}$ and $y_{\pm}$, which are sensitive to the Cabibbo-Kobayashi-Maskawa angle $\gamma$. The analysis exploits a sample of proton-proton collision data corresponding to 3.0 fb$^{-1}$ collected by the LHCb experiment. Measurements from CLEO-c of the variation of the strong-interaction phase of the $D$ decay over the Dalitz plot are used as inputs. The values of the parameters are found to be $x_+ = ( -7.7 \pm 2.4 \pm 1.0 \pm 0.4 )\times 10^{-2}$, $x_- = (2.5 \pm 2.5 \pm 1.0 \pm 0.5) \times 10^{-2}$, $y_+ = (-2.2 \pm 2.5 \pm 0.4 \pm 1.0)\times 10^{-2}$, and $y_- = (7.5 \pm 2.9 \pm 0.5 \pm 1.4) \times 10^{-2}$. The first, second, and third uncertainties are the statistical, the experimental systematic, and that associated with the precision of the strong-phase parameters. These are the most precise measurements of these observables and correspond to $\gamma = (62^{\,+15}_{\,-14})^\circ$, with a second solution at $\gamma \to \gamma + 180^\circ$, and $r_B = 0.080^{+ 0.019}_{-0.021}$, where $r_B$ is the ratio between the suppressed and favoured $B$ decay amplitudes.