Measurement of $CP$ Violation in $B \rightarrow D K^{*}$ Decays with the LHCb Experiment

This thesis reports an analysis of 3.0 fb$^{-1}$ of $pp$ collision data collected by the LHCb experiment in 2011 and 2012. Decays of neutral $B$ mesons to neutral $D$ mesons and excited neutral kaons are reconstructed, because of their sensitivity to the $CP$-violating phase of the CKM matrix, $\gamma$. The neutral $D$ meson is reconstructed in four separate final states. These are $K^- \pi^+$, $K^+ \pi^-$, $K^+ K^-$ and $\pi^+\pi^-$. Several observables are defined, in terms of the partial widths of $B\to D K^*$ decays, that are sensitive to $\gamma$ and other, $CP$-conserving, parameters of the decays of the $B$ and $D$ mesons. The measured results include the $CP$ asymmetry between the partial widths of $\bar {B}^0 \to D(K^+K^-) \bar {K}^{*0}$ and $B^0 \to D (K^+ K^-) K^{*0}$ decays, which is found to be \begin{equation*} {\cal A}_d^{KK} = -0.198 \, ^{+0.144}_{-0.145} \, ^{+0.019}_{-0.020}. \end{equation*} In addition, the Cabibbo-suppressed $B^0 \to D(K^- \pi^+) K^{*0}$ and $ \bar {B}^0 \to D(K^+ \pi^-) \bar {K}^{*0}$ decays are analysed for the first time with data collected by LHCb. The ratio of the partial widths of the Cabibbo-suppressed $B^0 \to D (K^- \pi^+) K ^{*0}$ decay to the Cabibbo-favoured $B^0 \to D(K^+ \pi^-) K^{*0}$ decay is measured, as well as the analogous quantity corresponding to $ \bar {B}^0$ decays. These are found to be \begin{equation*} {\cal R}_d^+ = 0.057 \, ^{+0.029}_{-0.027} \, ^{+0.009}_{-0.012} \space \space \space \space \space \space \space \space \space and \space \space \space \space \space \space \space \space \space {\cal R}_d^- = 0.056 \, ^{+0.032}_{-0.030} \, ^{+0.009}_{-0.012}, \end{equation*} respectively. Finally, the ratio of the magnitudes of the amplitudes corresponding to $B^0 \to D^0 K^{*0}$ and $B^0 \to \bar{D}^0 K ^{*0}$ decays is found to be \begin{equation*} r_B = 0.230 \, ^{+0.063}_{-0.045} \end{equation*} at a confidence level of 68.3 % and the first constraints on $\gamma$ from $B_d \to DK^*$ decays are made.