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

Mixing is the time-dependent phenomenon of a neutral meson (in this case charm meson $D^0$) changing into its anti-particle ($\bar{D}^0$) and vice versa. This occurs because the mass eigenstates, denoted $D_1$ and $D_2$ are linear combinations of the flavour eigenstates $D^0$ and $\bar{D}^0$. Mixing is governed by two parameters $x$ and $y$ defined as: $x\equiv(m_1-m_2)/\Gamma$ and $y\equiv(\Gamma_1-\Gamma_2)/(2\Gamma)$ where $\Gamma$ is the average decay width. CP-violation can occur in mixing or in the interference between mixing and decay. The CP-violation parameters $|q/p|$ and $\phi$ describe the superposition of the flavour eigenstates and the mass eigenstates: $\vert{D_{1,2}}=p\vert{D^0}\pm q\vert{\bar{D}^0}$. The self-conjugate decay $D^0\to K^0_S\pi^+\pi^-$ offers direct access to the mixing and CP-violation parameters through a time and phase-space dependent fit to the Dalitz variables and decay-time of this decay. This thesis reports a measurement of the mixing and \CP-violation parameters using data collected at the LHCb experiment in the Run 2 data-taking period in 2016-2018, corresponding to an integrated luminosity of 6~fb$^{-1}$. This analysis uses $D^0$ mesons originating from semi-leptonic $B$ meson decays. The $D^0\to K^0_S\pi^+\pi^-$ decay is modelled by expressing the three-body decay as the superposition of successive two-body decays through intermediate resonances. The blinded mixing parameters are found to be: \begin{equation*} \begin{split} x &= (x.xx \pm 0.86_{\text{stat}} \pm 0.39_{\text{syst}} \pm 0.24_{\text{model}})\times 10^{-3} \\ y &= (y.yy \pm 0.76_{\text{stat}} \pm 0.59_{\text{syst}} \pm 0.26_{\text{model}})\times 10^{-3} \end{split} \end{equation*} where the uncertainties are statistical, systematic and from the choice of amplitude model. The \CP-violation parameters are expressed in terms of $\Delta x$ and $\Delta y$ which are defined as the difference in mixing parameters measured for $D^0$ and $\bar{D}^0$: \begin{equation*} \begin{split} \Delta x &= (0.00 \pm 0.59) \times 10^{-3} \\ \Delta y &= (0.00 \pm 0.51) \times 10^{-3} \\ \end{split} \end{equation*} the uncertainties are currently statistical only and the results are blind.