Test of Lepton Flavour Universality using the $B_s^0\to D_s^- \tau^+ \nu_\tau$ with 3 prongs $\tau^+$ decays and validation of the new opto-electronics for the RICH Upgrade at the LHCb experiment

The Standard Model (SM) of particle physics assumes that the couplings between leptons and the electroweak gauge bosons are independent of the lepton flavour up to a correction due to the mass. This property is known as Lepton Flavour Universality (LFU). Ideal laboratories to test the LFU are the $b$-hadron semileptonic decays which can be studied through the ratios of branching fractions between decays with the $\tau$ lepton and the ones with the $\mu$ lepton in the final state: \[R(H_c)= \frac{\mathcal{B}(H_b\to H_c\tau\nu)}{\mathcal{B}(H_b\to H_c\mu\nu)},\] with $H_b$ and $H_c$ the $b-$hadron which originates the decay and the $c-$hadron produced in the decay respectively. Experimental results on LFU tests have been obtained by Belle, BaBar and LHCb collaborations and show tension with the SM prediction of about 3.4$\sigma$ when considering the combination of the measurement of $R(D)$ and $R(D^*)$. Exploiting the abundance of the $b$-hadron produced in the LHCb environment and the features of the detector, that allow to reconstruct the $b$-hadron decay vertex with high precision and to perform particle identification, this thesis documents the preliminary studies for the measurement of $R(D_s)$ defined as: \[R(D_s)= \frac{\mathcal{B}(B^0_s\to D^-_s\tau^+\nu_\tau)}{\mathcal{B}(B^0_s\to D^-_s\mu^+\nu_\mu)},\] with using the $\tau$ decaying in 3 charged pions. The data sample used for the studies corresponds to an integrated luminosity of 2 fb$^{-1}$ of proton proton collision events at a centre-of-mass energy of 8 TeV collected by the LHCb experiment in 2012. Given the request of high precision measurements to further test the SM in many other sectors, the LHCb experiment is currently being upgraded to be able to operate at a luminosity of about 2$\times 10^{33}$cm$^{-2}$s$^{-1}$ from the start of Run3 in 2022. To cope with the challenge of a five-fold increased instantaneous luminosity, and the need to read-out the full LHCb detector at a rate of 40 MHz, the photon detectors and the electronics chain of both the Ring Imaging CHerenkov (RICH) detectors have been replaced. This thesis also describes the quality assurance and test protocols on the new opto-electronics and reports of the commissioning activities.