A measurement of D-mixing in wrong sign $D^{0}\to K^+ \pi^- \pi^+ \pi^-$ decays to provide input to a model-independent determination of the CKM phase $\gamma$

A measurement of the time dependent ratio of $D^{0}\to K^+ \pi^- \pi^+ \pi^-$ to $D^{0}\to K^- \pi^+ \pi^- \pi^+$ decays is sensitive to both D-mixing and the interference between doubly Cabibbo suppressed (DCS) and Cabibbo favoured (CF) $D^{0}\to K \pi \pi \pi$ amplitudes. Such a measurement is made using 1.0 $\mathrm{fb}^{-1}$ of data collected by the LHCb experiment at a proton-proton centre of mass collision energy of $\sqrt{s} = 7$ TeV. The ratio of DCS to CF amplitudes is measured to be, $r_D^{K3\pi} = 0.0548 \pm 0.0012$. The interference between DCS and CF amplitudes is described by the complex interference parameter $\mathcal{Z}^{K3\pi}$. A combination of results from LHCb and CLEO-c gives the following constraints: $\mathcal{Re}\mathcal{Z}^{K3\pi}= -0.135^{+0.095}_{-0.105}$, $\mathcal{Im}\mathcal{Z}^{K3\pi} = -0.26^{+0.19}_{-0.16}$. The probability of the data being a statistical fluctuation of the no-mixing hypothesis is $2.5\times10^{-6}$, corresponding to a significance of $4.7\sigma$ - this is the first evidence of D-mixing in this decay mode. A model-independent measurement of the CKM phase $\gamma$ in $B\to DK, D \to f$ decays requires external input for $r^{f}_D$ and $\mathcal{Z}^{f}$. Previously such information was only available from the charm threshold at an $e^+e^-$ collider. Presented in this thesis is a new method that only requires input from charm mixing. The novel approach is only possible when the final state of the $D$ decay is multi-body, and therefore the approach is demonstrated with a simulation study of the decay mode $D\to K\pi\pi\pi$. The performance of the method is evaluated with input from both charm mixing, and the charm threshold. Individually, both methods give useful constraints, but if inputs from both are combined, the method becomes highly competitive with measurements of $\gamma$ in other decay modes. With assumptions about the resonant substructure of the DCS amplitude, it is estimated that $\gamma$ can be measured to within $12^{\circ}$ using existing data and $4^{\circ}$ degrees for the LHCb-upgrade.