Measurements of Angular Observables of $B^0\to K^{*0} \mu^+ \mu^-$ and $B^0 \to K^{*0} e^+ e^-$ Decays and the Upgrade of LHCb

The LHCb experiment at the Large Hadron Collider is tailored to search for indirect evidences of physics beyond the Standard Model, mainly by studying $CP$ violation and rare decays of charm and bottom hadrons. Despite the great success of LHCb, several key measurements will be limited by statistical uncertainties even after seven years of data taking. In order to increase the annual event yields, the LHCb detector will undergo a major upgrade during the Long Shutdown II, which will require the replacement of several subdetectors. The tracking system, in particular, will be completely redesigned in order to deal with the higher instantaneous luminosity and guarantee a full read out at the proton-proton collision rate of 40 $\rm{MHz}$. The first part of this thesis describes the design and testing of a new tracking detector, the Upstream Tracker. As its predecessor, the Upstream Tracker consists of silicon microstrip sensors. Its design involves two key elements that require dedicated measurements on prototype sensors before production: the adoption of an embedded pitch adapter to connect the silicon microstrips to the readout electronics and the configuration of the biasing scheme. Tests on prototype sensors were performed with beams of particles at the Super Proton Synchrotron in the past years and allowed to characterise the long-term performances of the sensors in terms of charge collection efficiency, cross talk, and radiation hardness. These studies resulted in the final design of the outermost sensors of the Upstream Tracker. One of the most promising areas of research of flavour physics is the study of rare decays of bottom hadrons and, in particular, of those proceeding through a $b \to s \ell^+ \ell^-$ transition, with $\ell = e,\mu$. These decays are particularly powerful tools to probe New Physics contributions, since the latter can result in sizeable deviations from the Standard Model predictions. In recent years, hints of deviations were observed in the angular distribution of $B^0\to K^{*0} \mu^+ \mu^-$ decays, $e.g.$ the $P_5^\prime$ anomaly, as well as in branching fraction ratios between muons and electrons, like $R_K$ and $R_{K^*}$. These observations strongly increased the interest toward New Physics models in which lepton flavour universality is violated. Breaking of lepton flavour universality through New Physics effects can be measured by comparing the values of the $A_{FB}$, $S$, and $P^\prime$ observables of $B^0\to K^{*0} \mu^+ \mu^-$ and $B^0 \to K^{*0} e^+ e^-$ decays. The second part of this thesis describes the first measurement of the $\Delta A_{FB} = A_{FB}^e - A_{FB}^\mu$ and $\Delta S_i = S_i^e - S_i^\mu$ observables, with $i=4,5,7,8$, using the dataset collected by LHCb from 2010 to 2012. These observables are expected to be compatible with zero in the Standard Model, but can significantly deviate from this value in the event of New Physics. The analysis described in this thesis is performed using a counting method procedure, which is based on the symmetry properties of the $CP$-average differential decay width of the $B^0 \to K^{\star 0} \ell^+ \ell^-$ decay. This approach is very robust, even with limited statistics, and is independent of the angular distribution of background contributions. The measurement is performed in two bins of $q^2$ (that is, of the dilepton invariant mass squared), in the low $q^2$ region from 0.1 $\rm {GeV}^2/c^4$ to 1.1 $\rm {GeV}^2/c^4$, and in the central $q^2$ region from 1.1 $\rm {GeV}^2/c^4$ to 7.0 $\rm {GeV}^2/c^4$. The results are compatible with the Standard Model predictions and are dominated by statistical uncertainties. The analysis presented here shows the capabilities of the LHCb experiment in measuring the angular observables of $B^0 \to K^{\star 0} \ell^+ \ell^-$ decays and provides a baseline for upcoming measurements, which will include the dataset collected since 2012.