Measurement of the strong-phase difference between $D^0$ and $\kern 0.2em\overline{\kern -0.2em D}{}^0$ decays to $K^0_SK^+K^-$ at CLEO-c and a determination of observables related to $\mathcal{CP}$ violation in $B^\pm \to D K^\pm$ decays at LHCb

A central goal of flavour physics is a precise determination of the elements of the CKM matrix, which quantifies the strength of charged-current weak interactions between quarks. Of particular interest is the angle $\gamma$ in the `$b-d$´ unitarity triangle parameterisation of the CKM matrix. One of the most promising methods to determine $\gamma$ directly is to measure $\mathcal{CP}$ violation in interfering $B^\pm \to D K^\pm$ decays, where $D$ indicates a coherent superposition of $D^0$ and $\kern 0.2em\overline{\kern -0.2em D}{}^0$, both of which decay to the same final state. When using this method it is essential to determine the hadronic decay parameters of the $D$ precisely in order to reduce the systematic uncertainties on the measurement of $\gamma$. One such parameter is the strong-phase difference between $D^0$ and $\kern 0.2em\overline{\kern -0.2em D}{}^0$ decays, which must be accurately known across the entire kinematic phase space. In this thesis we present measurements related to the determination of $\gamma$ at both the CLEO-c experiment at Cornell University and the LHCb experiment at CERN. Firstly, we describe a model-independent determination of the $D \to K^0_SK^+K^-$ strong-phase difference using 818 pb$^{-1}$ of quantum-correlated $D^0 \kern 0.2em\overline{\kern -0.2em D}{}^0$ data collected by CLEO-c at the $\psi(3770)$ resonance. We reconstruct $D \to K^0_SK^+K^-$ decays tagged with a variety of final states. By studying these decays we determine the weighted cosine and sine of the strong-phase difference in bins across the Dalitz plane. We run simulations to estimate the impact of these measurements on a determination of $\gamma$ using $B^\pm \to (K^0_SK^+K^-)_D K^\pm$ decays. The resulting uncertainty on $\gamma$ due to the CLEO-c inputs is between 3.2$^\circ$ and 3.9$^\circ$, depending on how the Dalitz plane is binned. Furthermore, we present a model-independent measurement of the $\mathcal{CP}$ content of the decay $D^0 \to K^0_SK^+K^-$ in the kinematic region of the $\phi \to K^+K^-$ resonance. The fraction of $\mathcal{CP}$-odd events in this region is 0.76 or higher at the 90 % C.L. We also present an analysis of data recorded by LHCb in 2010, corresponding to an integrated luminosity of 36.5 pb$^{-1}$. We reconstruct the decays $B^\pm \to (K^\pm\pi^\mp)_D h^\pm$ and $B^\pm \to (K^+ K^-)_D h^\pm$, where $h^\pm$ indicates either $K^\pm$ or $\pi^\pm$. Although there are not enough events in this dataset to measure $\gamma$, we are able to measure other observables related to $\mathcal{CP}$ violation in the $B^\pm \to D h^\pm$ system. We measure $\mathcal{B}^{DK}_{\mathrm{Fav}}/\mathcal{B}^{D\pi}_{\mathrm{Fav}}$, the ratio of the branching fraction of $B^\pm \to (K^\pm\pi^\mp)_D K^\pm$ to that of $B^\pm \to (K^\pm\pi^\mp)_D \pi^\pm$, to be $0.066 \pm 0.005 \pm 0.004$, and $\mathcal{B}^{DK}_{\mathcal{CP}}/\mathcal{B}^{D\pi}_{\mathcal{CP}}$, the ratio of the branching fraction of $B^\pm \to (K^+ K^-)_D K^\pm$ to that of $B^\pm \to (K^+ K^-)_D \pi^\pm$, to be $0.093 \pm 0.019 \pm 0.005$. We determine several $\mathcal{CP}$ asymmetries: $A_{\mathcal{CP}+}^{DK}$, the $\mathcal{CP}$ asymmetry in $B^\pm \to (K^+ K^-)_D K^\pm$ decays, is measured as $0.06 \pm 0.17 \pm 0.07$; $A_{\mathcal{CP}+}^{D\pi}$, the $\mathcal{CP}$ asymmetry in $B^\pm \to (K^+ K^-)_D \pi^\pm$ decays, is found to be $0.009 \pm 0.042 \pm 0.011$; and $A_{\mathrm{Fav}}^{DK}$, the $\mathcal{CP}$ asymmetry in $B^\pm \to (K^\pm\pi^\mp)_D K^\pm$ decays, is measured as $-0.109 \pm 0.085 \pm 0.019$. Finally we calculate $R_{\mathcal{CP}+}$, the ratio of the branching fraction of $B^\pm \to (K^+ K^-)_D K^\pm$ to that of $B^\pm \to (K^\pm\pi^\mp)_D K^\pm$, to be $1.41 \pm 0.31 \pm 0.11$. These results indicate that LHCb is in a strong position to make a world-leading measurement of $\gamma$ with a larger data sample