Search for flavour anomalies at LHCb: decay-time-dependent CP violation in $B_s^0\to(K^+\pi^-)(K^-\pi^+)$ and Lepton Universality in $\overline{B}^0\to D^{(*)+}l\overline{\nu_{l}}$

The Standard Model (SM) of Particle Physics is a very successful theory, but it leaves some important questions unanswered. One long-standing mystery is the so called flavour structure, consisting on the specific arrangement of fermions in three families and the connections among them. Several New Physics (NP) models predict modifications to the flavour structure and consequent deviations from the expected behaviour of sub-atomic processes. The LHCb experiment, at CERN, searches for NP by measuring the properties of B mesons decays, using B mesons produced in p-p collisions at the LHC. This thesis presents two measurements performed in LHCb that allow to test the flavour structure of the SM in different ways. The first and main analysis corresponds to the World's first measurement of the Charge-Partity (\CP)-violating phase $\phi_s^{d\overline{d}}$, using $B^0_s \rightarrow (K^+\pi^-)(K^-\pi^+)$ decays. The analysis employs data collected by LHCb in 2011 and 2012. The physical motivation is the \CP problem, corresponding to the inconsistency between the huge matter/anti-matter asymmetry in the Universe and the small \CP asymmetry that the SM can accommodate (embedded in the flavour structure). The $B_s^0\to K^{*0}(\to K^+\pi^-)\overline{K}^{*0}(\to K^-\pi^+)$ decay is forbidden at tree level in the SM, only proceeding via loop (penguin) diagrams. This suppression allows potential NP contributions to play a significant role. The $\phi_s^{d\overline{d}}$ phase, predicted to be approximately 0 in the SM, arises in the interference between the $B^0_s/ \bar{B^0_s}$ oscillation and the decay of either a $B^0_s$ or a $\bar{B^0_s}$ to the $K^{*0}\overline{K}^{*0}$ final state. To increase the available statistics, not only the $K^{*0}(\overline{K}^{*0})$ resonance is inspected, but also several scalar, vector and tensor $K\pi$ components present in a two-dimensional $K\pi$ mass window of 750 to 1600 \MeVcc. A total of 19 amplitudes are studied. Given the current level of sensitivity, the same $\phi_s^{d\overline{d}}$ phase is used for all the states. A full amplitude analysis is done via an un-binned fit to the multi-dimensional decay distribution on the $B^0_s$ decay-time, three angular variables and the two $K\pi$ invariant masses. The \CP-violating phase is measured to be $\phi_s^{d\overline{d}}= -0.10 \pm 0.13 \pm 0.14$rad, where the first uncertainty is statistical and the second one is systematic. It is compatible with the SM prediction, within uncertainties. Many other amplitude-parameters are determined in addition. In particular, the longitudinal-polarisation fraction of the $B_s^0\to K^{*0}\overline{K}^{*0}$ decay is found to be $f_{\rm L}=0.208\pm0.032\pm0.046$, which confirms the previously measured low value and deviates from the existent expectations based on the SM. The second analysis aims at testing the Lepton Universality. This property of the SM consists on the electro-weak couplings being the same for the three lepton families. Several deviations from this behaviour have been recently observed, the largest one corresponding to the combined measurement of $R(D)$ and $R(D^*)$ and being of 4.1 standard deviations. The $R$ observables are defined as $R(D^{(*)})=\mathcal{B}(B\to D^{(*)}\tau\bar{\nu}_{\tau})/\mathcal{B}(B\to D^{(*)}\mu\bar{\nu}_{\mu})$, where $D$ is $D^0$ or $D^+$ and $D^{*}$ is $D^{*0}$ or $D^{*+}$. The analysis presented in this work corresponds to the first LHCb measurement of $R(D)$ using $D^+$, and is restricted to the case $\tau\to \mu\nu_{\tau}\bar{\nu}_{\mu}$ and $D^+\to K^-\pi^+\pi^+$. A data sample collected by LHCb in 2015 is used. The measurement is done through a binned, three-dimensional, template-based fit to filtered candidates with the visible final state $K^-\pi^+\pi^+\mu^-$. Contributions from $D^{*+}\to D^+\pi^0$ to the same visible state, when the $\pi^0$ is not reconstructed, allow to measure $R(D^*)$ in combination with $R(D)$. After filtering, a total yield of about 800 000 $D^+$ candidates is found. The thesis reports the work done in this analysis, which will be continued in the future. To improve the sensitivity, the 2015 data sample will be combined for the final result with the data collected in 2016.