Measurement of $CP$ violation and constraints on the CKM angle $\gamma$ in $B^{\pm }\to D{K}^{\pm }$ with $D\to K_S^0{\pi }^{+}{\pi }^{-}$ decays

A model-dependent amplitude analysis of $B^{\pm }\to D{K}^{\pm }$ with $D\to K_S^0{\pi }^{+}{\pi }^{-}$ decays is performed using proton-proton collision data, corresponding to an integrated luminosity of $1$ fb${}^{-1}$, recorded by LHCb at a centre-of-mass energy of $7$ TeV in $2011$. Values of the $CP$ violation observables $x_{\pm }$ and $y_{\pm }$, which are sensitive to the CKM angle $\gamma$, are measured to be \begin{align*} x_- &= +0.027 \pm 0.044 ^{+0.010}{-0.008} \pm 0.001, \ y- &= +0.013 \pm 0.048 ^{+0.009}{-0.007} \pm 0.003, \ x+ &= -0.084 \pm 0.045 \pm 0.009 \pm 0.005, \ y_+ &= -0.032 \pm 0.048 ^{+0.010}_{-0.009} \pm 0.008, \end{align*} where the first uncertainty is statistical, the second systematic and the third arises from the uncertainty of the $D\to K_S^0{\pi }^{+}{\pi }^{-}$ amplitude model. The value of $\gamma$ is determined to be $(84_{-42}^{+49}{)}^{\circ }$, including all sources of uncertainty. Neutral $D$ meson mixing is found to have negligible effect.