Study of the $B^0\to \rho(770)^0 K^*(892)^0$ decay with an amplitude analysis of $B^0\to (\pi^+\pi^-) (K^+\pi^-)$ decays

This thesis is devoted to the study of the $B^0\to \rho(770)^0 K^*(892)^0$ decay mode for which 3 fb$^{-1}$ of proton-proton collision data collected by the LHCb experiment at centre-of-mass energies of 7 and 8 TeV were analysed. The details of an amplitude analysis of ${B^0\to (\pi^+\pi^-) (K^+\pi^-)}$ decays performed in the two-body invariant mass regions corresponding to \mbox{${300 < m(\pi^+\pi^-)<1100}$ MeV/$c^2$} and \mbox{${750 < m(K^+\pi^-)<1200}$ MeV/$c^2$} are documented and its results are discussed. Particular emphasis is placed in the measurements of the \ensuremath{C\!P} averages and asymmetries for the magnitudes and phase differences of the contributing amplitudes. The presented results correspond to the first measurements of most of these observables. In particular, the \ensuremath{C\!P}-averaged longitudinal polarisation fractions of the vector-vector modes are found to be \mbox{$\tilde{f}^0_{\rho K^{*}} = 0.164 \pm 0.015 \pm 0.022$} and \mbox{$\tilde{f}^0_{\omega K^{*}} = 0.68 \pm 0.17 \pm 0.16$}, and their \ensuremath{C\!P} asymmetries, \mbox{$\mathcal{A}^0_{\rho K^{*}} = -0.62 \pm 0.09 \pm 0.09$} and \mbox{$\mathcal{A}^0_{\omega K^{*}} = -0.13 \pm 0.27 \pm 0.13$}, where the first uncertainty is statistical and the second systematic. Consequently, the first observation of direct \ensuremath{C\!P} violation in amplitude analyses of vector-vector decays is reported.