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

A model-dependent amplitude analysis of $B^{pm} rightarrow D K^{pm}$ with $D rightarrow K_S^0 pi^+ pi^-$ 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 $D rightarrow K_S^0 pi^+ pi^-$ amplitude model. The value of $gamma$ is determined to be $(84^{+49}_{-42})^circ$, including all sources of uncertainty. Neutral $D$ meson mixing is found to have negligible effect.