Observation of $CP$ violation in $B^\pm \to D K^\pm$ decays

An accurate determination of the angle $\gamma$ of the Unitary Triangle is one of the most important goals of the LHCb experiment. The LHCb detector is a single-arm spectrometer at the LHC, optimised for beauty and charm flavour physics. As the angle $\gamma$ is the least experimentally constrained parameter of the Unitary Triangle, its precise experimental determination can be used to test the validity of the Standard Model. The Unitary Triangle phase $\gamma$ can be extracted in $B \rightarrow DK$ decays at tree-level, exploiting the interference between $b \rightarrow c(\bar{u}s)$ and $b \rightarrow u(\bar{c}s)$ transitions. This interference is sensitive to $\gamma$ and can give measurable charge asymmetries. In particular, $\gamma \neq 0$ is required to produce direct $CP$ violation in $B$ decays and this is the only $CP$-violating mechanism for the decay of charged $B^\pm$ mesons. In this thesis, an analysis of $CP$ violation in $B^{\pm} \rightarrow DK^{\pm}$ and $B^{\pm} \rightarrow D\pi^{\pm}$ decays is presented, where the $D$ meson is reconstructed in the two-body final states: $K^{\pm}\pi^{\mp}$, $K^+K^-$, $\pi^+\pi^-$ and $\pi^{\pm}K^{\mp}$. The analysis uses the full 2011 LHCb dataset of 1 $\textrm{fb}^{-1}$, collected from $pp$ collisions at $\sqrt{s} = 7$~TeV. Several $CP$-related quantities, e.g the ratio of $B \rightarrow DK$ and $B \rightarrow D\pi$ branching fractions and their charge asymmetries, are measured via a simultaneous fit to the invariant mass distributions of the modes considered. The suppressed $B^{\pm} \rightarrow DK^{\pm}$ mode is observed for the first time with $\approx 10\sigma$ significance. Once all measurements are combined, direct $CP$ violation is established in $B^{\pm}$ decays with a total significance of $5.8\sigma$.