Studies of $\mathrm{D}^0\to\mathrm{K}^0_{\scriptscriptstyle\rm S}\mathrm{h}^{+}\mathrm{h}'^{-}$ decays at the LHCb experiment

This thesis documents two studies of the neutral charm meson system using the LHCb detector, and gives an overview of the numerous changes made to the LHCb software trigger in advance of Run~2 of the LHC. In the first analysis, amplitude models are applied to studies of the resonance structure in $\mathrm{D}^0\to\mathrm{K}^0_{\scriptscriptstyle\rm S}\mathrm{K}^{-}\pi^{+}$ and $\mathrm{D}^0\to\mathrm{K}^0_{\scriptscriptstyle\rm S}\mathrm{K}^{+}\pi^{-}$ decays using proton-proton collision data, corresponding to an integrated luminosity of $3.0\mbox{ fb}^{-1}$, collected during Run~1 of the LHC. Relative magnitude and phase information is determined, and coherence factors and related observables are computed for both the whole phase space and a restricted region of $100\mathrm{\,Me\kern -0.1em V\!/}c^2$ around the $\mathrm{K}^{*}(892)^{\pm}$ resonance. Two formulations for the $\mbox{$\mathrm{K}\pi$ $S$-wave}$ are used, both of which give a good description of the data. The ratio of branching fractions $\mathcal{B}\left(\mathrm{D}^0\to\mathrm{K}^0_{\scriptscriptstyle\rm S}\mathrm{K}^{+}\pi^{-}\right)/\mathcal{B}\left(\mathrm{D}^0\to\mathrm{K}^0_{\scriptscriptstyle\rm S}\mathrm{K}^{-}\pi^{+}\right)$ is measured to be $0.655\pm0.004\mathrm{\,(stat)}\pm0.006\mathrm{\,(syst)}$ over the full phase space and $0.370\pm0.003\mathrm{\,(stat)}\pm0.012\mathrm{\,(syst)}$ in the restricted region. A search for $C\!P$ violation is performed using the amplitude models and no significant effect is found. Predictions from $\mathrm{SU}(3)$ flavour symmetry for $\mathrm{K}^{*}(892)\mathrm{K}$ amplitudes of different charges are compared with the amplitude model results, and marginal agreement is found. The second study estimates the sensitivity to $\mathrm{D}^0$--$\overline{\mathrm{D}}^0$ mixing and indirect $C\!P$ violation parameters that can be achieved using a model-independent technique and the samples of $\mathrm{D}^0\to\mathrm{K}^0_{\scriptscriptstyle\rm S}\mathrm{K}^+\mathrm{K}^-$ decays recorded by LHCb in Run~1 and Run~2 of the LHC. These studies show that constraints on these parameters could be significantly improved by an analysis of the anticipated Run~2 dataset.