Measurement of the mixing parameters of neutral charm mesons and search for indirect $CP$ violation with $D^0 \to K^0_S \pi^+ \pi^-$ decays at LHCb

The hadronic decay $D^0 \to K^0_S \pi^+ \pi^-$ provides direct access to the measurement of the mixing parameters of the neutral charm meson system and allows to test for indirect $CP$ violation. Mixing is a time-dependent phenomenon for which the time evolution of the transition amplitude of a $D^0 \, (\bar{D}^0)$ decay to the final state $K^0_S\pi^+\pi^-$ has to be considered. The parameters driving those time-dependent oscillations are $x \equiv (m_1-m_2)/\Gamma$ and $y \equiv (\Gamma_1-\Gamma_2)/(2\Gamma)$. The $CP$ violation parameters $|q/p|$ and $\phi=\arg(q,p)$ describe the superposition of the flavour eigenstates $D^0$ and $\bar{D}^0$ and of the physical eigenstates $D_1$ and $D_2$, $|D_{1,2}\rangle = p |{D^0}\rangle \pm q |{\bar{D}^0}\rangle$. By measuring the time- and phase-space dependent distribution of $D^0 \to K^0_S \pi^+ \pi^-$ decays, the mixing parameters can be extracted and a search for indirect $CP$ violation can be performed. This thesis reports a measurement of the mixing parameters and the preparations of a measurement of the $CP$ violation parameters on data collected with the LHCb experiment in 2011 and 2012, corresponding to an integrated luminosity of $3\, \mathrm{fb}^{-1}$. The $D^0$ and $\bar{D}^0$ mesons are required to originate from a semileptonic decay of a $B$ meson. The parameters of interest are extracted from a fit in $D^0$ decay time and the Dalitz variables, $m^2(K^0_S\pi^-)$ and $m^2(K^0_S\pi^+)$. The phase-space distribution of $D^0 \to K^0_S \pi^+ \pi^-$ decays is modelled by expressing the three-body decay as a succession of two-body decays. The decay amplitude of a $D^0$ or $\bar{D}^0$ meson into the $K^0_S \pi^\pm \pi^\mp$ final state is a superposition of all possible intermediate resonances and the single resonances interfere with each other across the phase-space. The blinded $D^0-\bar{D}^0$ mixing parameters are found to be \begin{align*} x &= (-4.76 \pm 0.22_{\rm \, stat.} \pm 0.05_{\rm \, syst.} + 0.08_{\rm \, model}) \%,\\ y &= (-4.13 \pm 0.19_{\rm \, stat.} \pm 0.07_{\rm \, syst.} + 0.05_{\rm \, model})\%, \end{align*} where the uncertainties are statistical, systematic and model-dependent.