$C\!P$ violation in $D^0 \to K^+ K^-$ and $D^0 \to \pi^+ \pi^-$ decays and lepton-flavour universality test with the decay $B^0 \to D^{*-} \tau^+ \nu_\tau$

The LHCb experiment has been designed to exploit the potential of heavy-flavour production in highly energetic $pp$ collisions at the Large Hadron Collider, in order to look for indirect signs of physics beyond the Standard Model of particle physics. The experiment has been performing a number of measurements with a plethora of physics observables, notably including $C\!P$-violating asymmetries and lepton-flavour universality probes. This thesis presents two distinct measurements performed using LHCb data. The first is that of the ratio of branching fractions $\mathcal{R}(D^*) \equiv \mathcal{B}(B^0 \to D^{*-} \tau^+ \nu_\tau) / \mathcal{B}(B^0 \to D^{*-} \ell^+ \nu_\ell) $, with the $\tau$ lepton decaying to final states containing three charged pions, performed using a data sample of $pp$ collisions at the centre-of-mass energies of 7 and 8 TeV, corresponding to 3 $\mathrm{fb}^{-1}$ of integrated luminosity. The result is $$ \mathcal{R}(D^*) = 0.291 \pm 0.019 \pm 0.026 \pm 0.013, $$ where the first uncertainty is statistical, the second systematic and the third due to the knowledge of the $B^0 \to D^{*-}\pi^+ \pi^- \pi^+$ branching fraction, as this decay is used in the analysis as a normalisation for the intermediate measurement of $\mathcal{B}(B^0 \to D^{*-} \tau^+ \nu_\tau)$. This corresponds to one of the most precise single measurements of $\mathcal{R}(D^*)$ and to the first performed with the three-prong decay of the $\tau$ lepton to date. The measured value of $\mathcal{R}(D^*)$ is compatible with previous determinations and with the Standard Model expectation. The second measurement presented in this thesis is that of the difference between the $C\!P$ asymmetries in $D^0 \to K^+ K^-$ and $D^0 \to \pi^+ \pi^-$ decays, performed using a data sample of $pp$ collisions at the centre-of-mass energy of 13 TeV, corresponding to 6 $\mathrm{fb}^{-1}$ of integrated luminosity. The value of $\Delta A_{C\!P} \equiv A_{C\!P}(K^+K^-) - A_{C\!P}(\pi^+\pi^-)$, measured by reconstructing $D^0$ mesons which originate from $D^{*+} \to D^0 \pi^+$ decays or from $B \to D^0 \mu \nu X$ decays, is $$ \Delta A_{C\!P} = ( -17.1 \pm 3.0 \pm 1.0 ) \cdot 10^{-4}, $$ where the first uncertainty is statistical and the second systematic. By combining this result with previous LHCb measurements based on Run-1 data, the value of $\Delta A_{C\!P}$ results to be $$\Delta A_{C\!P} = (-15.4 \pm 2.9) \cdot 10^{-4},$$ which differs from zero by 5.3 standard deviations. This is the first observation of $C\!P$ violation in the decay of a charm hadron.