The first double Dalitz plane analysis of $B^0 \to D K^+ \pi^-$ decays at LHCb

This thesis presents the first double Dalitz plane analysis of $B^0 \to D K^+ \pi^-$ decays at LHCb, utilising data collected by the LHCb experiment in Runs 1 and 2 of the LHC, corresponding to an integrated luminosity of $9 \text{fb}^{-1}$. This is the first ever use of this novel analysis method. In this analysis \D refers to an admixture of the states \Dz and \Dzb, reconstructed in the final states $D \to K_S^0 h^+ h^-$, $D \to h^+ h^-$, and $D \to h^+ h^- h^+ h^-$, where $h$ is either a kaon or pion. The primary goal of this analysis is to measure the CKM angle $\gamma$, the central value of which is currently blinded. The statistical uncertainty is determined to be $\sigma_\gamma = 6.8^\circ$ and the total systematic uncertainty is yet to be determined. The penultimate chapter of this thesis presents the first feasibility study of measurements of the branching fractions of $b \to s \nu \bar{\nu}$ decays at the FCC running electron-positron collisions at the $Z$ pole center of mass energy. This study shows that for $B^0 \to K^{*0} \nu \bar{\nu}$ and $B^0_s \to \phi \nu \bar{\nu}$ decays the branching fractions could be measured with a relative uncertainty of $\mathcal{O}(1\%)$, and $\mathcal{O}(3\%)$ and $\mathcal{O}(10\%)$ for $B^0 \to K_S^0 \nu \bar{\nu}$ and $\Lambda_b^0 \to \Lambda \nu \bar{\nu}$ decays, respectively.