Test of lepton flavour universality with the simultaneous measurement of $\mathcal{R}(D^+)$ and $\mathcal{R}(D^{\ast+})$ with $\tau^- \to \mu^- \nu_\tau \overline{\nu}_\mu$ decays at the LHCb experiment

In the Standard Model of particle physics, the coupling of the electroweak gauge bosons to the leptons is independent of the lepton flavour. This property, known as Lepton Flavour Universality, is an accidental symmetry of the Standard Model, which can be tested in semileptonic $b$-meson decays. The variables used to test the Lepton Flavour Universality hypothesis are ratios of branching fractions between decays with the $\tau$ lepton and the ones with the $\mu$ lepton in the final state: \begin{equation*} \mathcal{R}(H_c) = \frac{\mathcal{B}(B \to H_c \tau \nu)}{\mathcal{B}(B \to H_c \mu \nu)}, \end{equation*} with $H_c$ a charmed meson produced in the decay. Any sign of deviation with respect to the Standard Model predictions in these variables could be a clear sign of New Physics effects. A tension at the level of $3\sigma$ with respect to the Standard Model predictions has been observed in the combination of the measurements of $\mathcal{R}(D)$ and $\mathcal{R}(D^\ast)$ performed by the Belle, BaBar and \lhcb collaborations. At the time of writing of this thesis, no measurement of the $\mathcal{R}(D)$ parameter has been performed by any hadron collider experiment. This thesis reports a simultaneous measurement of the $\mathcal{R}(D^+)$ and $\mathcal{R}(D^{\ast+})$ parameters performed using $\overline{B}^0 \to D^{(\ast)+} \ell^- \overline{\nu}_\ell$ decays. This measurement exploits leptonic decays of the $\tau$ lepton, $\tau^- \to \mu^- \overline{\nu}_\mu \nu_\tau$, using a data sample corresponding to an integrated luminosity of $2.0~\rm{fb}^{-1}$ collected in proton-proton collisions at a centre-of-mass energy of 13\tev at the LHCb experiment during the 2015 and 2016 data taking years. All the steps of the analysis have been performed and the main systematic uncertainties have been studied. The values of the measured parameters remain still blinded and the analysis is in internal review within the LHCb collaboration. The expected uncertainties on the parameters of interest are given by \begin{eqnarray*} \mathcal{R}(D^+) &= xxx \pm 0.033 (\rm{stat.}) \pm 0.037 (\rm{syst.}),\\ \mathcal{R}(D^{\ast+}) &= xxx \pm 0.040 (\rm{stat.}) \pm 0.070 (\rm{syst.}). \end{eqnarray*}