Studies of the decays $D^{0}\rightarrow K^{\mp}\pi^{\pm}\pi^{\pm}\pi^{\mp}$ at CLEO-c and LHCb

This thesis describes two studies of the four-body decays of the neutral charm meson, $D^{0} \rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ and its doubly Cabibbo-suppressed counterpart $D^{0} \rightarrow K^{+}\pi^{-}\pi^{-}\pi^{+}$. The first analysis is a model-independent determination of parameters that characterise the phase space averaged interference between the two amplitudes associated with each of these decay modes. The analysis exploits quantum correlations in $D\bar{D}$ pairs produced from the $\psi(3770)$ resonance in data collected with the CLEO-c detector. The second half of this thesis describes studies of the resonant structure of these decay modes using proton-proton collision data corresponding to an integrated luminosity of $3.0fb^{-1}$ collected by the LHCb experiment. Studies of the favoured mode, $D^{0} \rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ , are the most precise studies of the amplitude to date and this data set is one of the largest samples of any decay mode to be studied using an amplitude analysis. The study of the suppressed mode, $D^{0} \rightarrow K^{+}\pi^{-}\pi^{-}\pi^{+}$, is the first study of resonance structure of this decay mode, and is also one of the few existing studies of the sub-structure of a doubly Cabibbo-suppressed amplitude. The largest contributions to both decay amplitudes are found to come from axial resonances, with decay modes $D^{0}\to{a}_1(1260)^{+}K^{-}$ and $D^{0}\to{K}_1(1270/1400)^{+}\pi^{-}$ being prominent in $D^{0} \rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ and $D^{0} \rightarrow K^{+}\pi^{-}\pi^{-}\pi^{+}$, respectively.