Study of the decays $B^0_{(s)} \rightarrow K^{*0} \bar{K}^{*0}$ and first observation of the $B_{s}^{0} \rightarrow \phi \bar{K}^{*0}$ decay at the LHCb experiment

This thesis describes the study of the decays $B^{0} \rightarrow K^{*0} \bar{K}^{*0}$ and $B_{s}^{0} \rightarrow K^{*0} \bar{K}^{*0}$ with $3 {\rm \, fb}^{-1}$ of data collected by the LHCb experiment, $1 {\rm \, fb}^{-1}$ in $2011$ at an energy $\sqrt{s} = 7 {\rm \, TeV}$ and $2 {\rm \, fb}^{-1}$ in $2012$ at $\sqrt{s} = 8 {\rm \, TeV}$; and the search for the $B_{s}^{0} \rightarrow \phi \bar{K}^{*0}$ decay with the data taken in $2011$. Flavour-untagged and time-integrated amplitude analyses of the $B^{0} \rightarrow K^{*0} \bar{K}^{*0}$ and $B_{s}^{0} \rightarrow K^{*0} \bar{K}^{*0}$ decays are reported. For the former, it is the first study at LHCb, and the large longitudinal polarisation obtained, $f_{L} = 0.707 \pm 0.053 \, ({\rm stat})$, confirms the previous measurement from the BaBar experiment. For the latter, an update of an earlier LHCb analysis is presented, corroborating the low longitudinal polarisation, $f_{L} = 0.233 \pm 0.030 \, ({\rm stat})$. Moreover, the ratio of branching ratios between the two modes, ${\cal B} (B^{0} \rightarrow K^{*0} \bar{K}^{*0}) / {\cal B} (B_{s}^{0} \rightarrow K^{*0} \bar{K}^{*0})$, is measured. In addition, as a result of the work presented here, the $B_{s}^{0} \rightarrow \phi \bar{K}^{*0}$ decay is observed for the first time, with a significance of $6.5\sigma$. The branching ratio ${\cal B} (B_{s}^{0} \rightarrow \phi \bar{K}^{*0})$ is measured and found to be compatible within $1 \sigma$ with theoretical predictions. An angular analysis of the $B_{s}^{0} \rightarrow \phi \bar{K}^{*0}$ decay is included. The small value obtained for the longitudinal polarisation fraction follows the trend of the $b \rightarrow s$ penguin decays. The result is smaller than the predictions.