Measurement of the $CP$-even Fraction of the $D^0 \rightarrow 2\pi^+ 2\pi^-$ Decay using Quantum Correlated ${D\bar{D}}$ Pairs at CLEO-c, and Real-time Alignment of the LHCb RICH optical Systems

This thesis covers three subjects, namely the measurement of the $C\!P$-even fraction of the $D^0\rightarrow 2\pi^+2\pi^-$ decay, the real-time alignment of the LHCb RICH mirror systems and the estimation of the sensitivity to the CKM angle $\gamma$ that can be obtained using $B^{\pm}\rightarrow D(\rightarrow 2\pi^+2\pi^-)K^{\pm}$ events at LHCb.\\ \\ The $C\!P$-even fraction $F_{4\pi}^{+}$ of the $D^0\rightarrow 2\pi^+2\pi^-$ decay is measured using a dataset corresponding to 818$\,pb^{-1}$ of quantum correlated $D\bar{D}$ decays produced in electron-positron collisions at the $\psi(3770)$ resonance collected by the CLEO-c experiment at Cornell University. In the analysis, one of the correlated $D$ mesons is reconstructed as $D\rightarrow 2\pi^+2\pi^-$ while the other $D$ meson is reconstructed as $D^0\rightarrow K^0_{S,L}\pi^+\pi^-$. Sensitivity to the $C\!P$-even fraction of $D^0\rightarrow 2\pi^+2\pi^-$ is obtained by determining the variation of yields over the $D^0\rightarrow K^0_{S,L}\pi^+\pi^-$ phase space, specifically the variation of yields between bins of the $D^0\rightarrow K^0_{S,L}\pi^+\pi^-$ phase space. The $C\!P$-even fraction is measured to be $F_{4\pi}^{+}= 0.755 \pm \, 0.050 \, \text{(stat)} \pm 0.029 \, \text{(syst)} $.\\ \\ The LHCb RICH mirror alignment procedure using proton-proton collision data is implemented in the LHCb online reconstruction framework for Run\,II of the LHC. In Run\,II, all LHCb subdetectors are aligned and calibrated in real-time, that is between the two high level trigger stages HLT1 and HLT2. This enables the reconstruction in the high level trigger to be identical to the offline reconstruction and the direct use of the trigger output for physics measurements. The RICH mirror alignment was implemented and commissioned throughout 2015 and 2016 and the procedure was improved at different points. This reduced the time needed to perform an alignment from several days in Run\,I to about 20$\,min$ in Run\,II. The information gathered by the frequent running of the alignment in 2016 is used to further the understanding of the alignment procedure as well as the understanding of the LHCb detector itself.\\ \\ The sensitivity to the CKM angle $\gamma$ that can be achieved at the LHCb experiment with $B^{\pm}\rightarrow D(\rightarrow 2\pi^+2\pi^-)K^{\pm}$ decays, is studied. The study is performed using the $B^{\pm}\rightarrow D(\rightarrow 2\pi^+2\pi^-)K^{\pm}$ event yield in 8$\,fb^{-1}$ of the proton-proton collision data expected to be recorded by LHCb by the end of Run\,II of the LHC. The $B^{\pm}\rightarrow D(\rightarrow 2\pi^+2\pi^-)K^{\pm}$ event yield of the data collected in 2016 is determined and extrapolated to the expected full luminosity of Run\,II. A series of pseudo-experiments is used to simulate the distribution of $B^{\pm}\rightarrow D(\rightarrow 2\pi^+2\pi^-)K^{\pm}$ events over the bins of the $D\rightarrow 2\pi^+2\pi^-$ phase space and a fit is performed to extract the resulting uncertainty on the CKM angle $\gamma$. The statistical uncertainty is determined to be 20$^\circ$ for the 2016 dataset, 10$^\circ$ for the full expected Run\,II dataset and 9$^\circ$ for the combined Run\,I and Run\,II dataset. The current best measurements of $\gamma$ from single analyses have a statistical uncertainty of $\approx 15^\circ$ while the combined uncertainty on $\gamma$ is $\approx 7^\circ$. The measurement of the angle $\gamma$ using binned $B^{\pm}\rightarrow D(\rightarrow 2\pi^+2\pi^-)K^{\pm}$ decays can thus significantly contribute to the overall constraint on $\gamma$.