Test of Lepton Flavour Universality using Bs semileptonic decays

The Large Hadron Collider beauty experiment (LHCb) focuses its studies on measurements of the decay properties of heavy-flavour hadrons, i.e. mesons or baryons containing charm ($c$) or beauty ($b$) quarks. LHCb is a single-arm spectrometer situated on the LHC ring with a forward angular acceptance, built this way to detect the majority of the $b\bar{b}$ quarks pair produced in the $pp$ collisions. One interesting topic LHCb has been studying is related to Lepton Flavour Universality of the Standard Model (LFU) testing. In case this symmetry is violated experimentally, this would be a clear sign of New Physics with contributions that favor couplings with one lepton family rather than the others. LFU tests can be performed in $b$-hadron decays by comparing branching fractions to final states with different lepton species and also by checking the angular distributions of the decays of interest. Experimental results on LFU tests performed by Belle, BaBar and LHCb show some tensions with respect to the SM. In particular, measurements of branching fraction ratios of decays involving $b \to c \ell^- \bar{\nu}_\ell $ transitions, as $R(D)$ and $R(D^*)$ show a limited compatibility with the SM predictions of about $3.08 \sigma$. Similarly, measurements of branching fraction ratios of decays involving $b \to s \ell^- \ell^+$ rare decays, show a compatibility with the SM predictions of about $2.5 \sigma$. To contribute in clarifying this interesting picture, LHCb is planning and currently performing measurements to test LFU using different $b$-hadrons decays produced at LHC, including different decay modes. This thesis focuses on the $R(D_s)$ measurement, defined as: $$ R(D_s) = \frac{{\cal B}(B_s^0 \rightarrow D_s^- \tau^+ \nu_{\tau})}{{\cal B}(B_s^0 \rightarrow D_s^- \mu^+ \nu_{\mu})}\, .$$ The $B_s^0 \rightarrow D_s^- \tau^+ \nu_{\tau}$ decay is reconstructed through the $\tau$ three-prong hadronic decay, $\tau^+ \rightarrow \pi^+ \pi^- \pi^+ (\pi^0) \nu_{\tau}$, while the $D_s^-$ meson is reconstructed through the $D_s^- \rightarrow K^+ K^- \pi^-$ decay chain, so the visible final state consists of six charged tracks. In order to achieve the most precise experimental result, the measurement is performed by introducing a normalization channel similar to the signal and redefining the ratio as: $$ R(D_s) = \frac{{\cal B}(B_s^0 \rightarrow D_s^- \tau^+ \nu_{\tau})}{{\cal B}({\rm norm})} \times \frac{{\cal B}({\rm norm})}{{\cal B}(B_s^0 \rightarrow D_s^- \mu^+ \nu_{\mu})} = {\cal K} \times \alpha \, . $$ Here the ratio $\alpha$ is completely determined by external inputs, i.e. measurements of branching fractions by independent analyses, while the ratio $\cal K$ is to be determined from the analysis of the signal and normalization channels as: $$ {\cal K} = \frac{N_{\rm sig}}{\epsilon_{\rm sig}} \frac{\epsilon_{\rm norm}}{N_{\rm norm}}\frac{1}{{\cal B}(\tau^+ \rightarrow \pi^+ \pi^- \pi^+ (\pi^0) \nu_{\tau}) \times {\cal B}(D_s^- \rightarrow K^+ K^- \pi^-)} \, .$$ Here, $N_{\rm sig}$ ($N_{\rm norm}$) and $\epsilon_{\rm sig}$ ($\epsilon_{\rm norm}$) represent, respectively, the signal (normalization) yield and selection efficiency. The main purpose of this thesis is to choose among possible normalization channels that are suitable for the $R(D_s)$ measurement. As a general requirement, the normalization and the signal channels should be reconstructed and selected in a similar way so that, when measuring the ratio ${\cal K}$, any systematic uncertainties related to the reconstruction and selection efficiency will be strongly reduced. The normalization channels studied in this thesis are $B_s^0 \rightarrow D_s^- \pi^+\pi^+\pi^-$ with $D_s^- \rightarrow K^+ K^- \pi^-$ and $B^0 \rightarrow D^- \pi^+\pi^+\pi^-$ with $D^- \rightarrow K^+ \pi^- \pi^-$. Chapter 1 is an introduction to the LHCb experiment and its apparatus. In Chapter 2, an overview of the LFU and of the experimental tests performed so far are presented, leading to the motivations for the $R(D_s)$ measurement, that is discussed in Chapter 3. Chapter 4 details several aspects of the analysis concerning the signal and normalization channels selection, the efficiency calculation and a quantitative evaluation of the main systematic uncertainties contributing to the $R(D_s)$ measurement. Final conclusions are drawn at the end.