Direct and Indirect Searches for New Physics in $b \to s \ell \ell$ Decays

The current era of particle physics is dominated by searches for physics beyond the Standard Model, which can be pursued with direct or indirect approaches. This dissertation investigates the features of rare semileptonic $b \to s \ell\ell$ transitions as probe of New Physics. Recent analyses of these decays have indicated an anomalous behavior in measurements of angular distributions of the decay $B^0 \to K^{*0} \mu^+ \mu^-$ and in lepton-flavour-universality observables. However, due to the limited understanding of hadronic uncertainties, the origin of some of these deviations is of difficult interpretation. These uncertainties, in fact, can mimic or camouflage New Physics effects limiting the discovery potential of measurements of $B^0 \to K^{*0} \mu^+ \mu^-$ decays. In order to solve this long-standing issue, a novel combined theory-experimental approach is presented. The proposed method is first applied to examine the prospects of an amplitude analysis of $B^0 \to K^{*0} \mu^+ \mu^-$ decays and then extended to the channel $B^0 \to K^{*0} e^+ e^-$ as a test of Lepton Flavour Universality. It is found that a separation between New Physics effects and hadronic uncertainties is possible and, if the current hints are confirmed, an early discovery of physics beyond the Standard Model can be achieved within the end of LHCb Run-II. Furthermore, the proposed method not only surpasses the sensitivity of previous approaches, but will also enable a deeper understanding of the nature of New Physics. Moreover, sensitivity studies, including all the experimental effects, have been performed investigating the expected significance of the proposed measurement with the current data set collected by the LHCb experiment. Finally, the study of $B^+ \to K^+ \mu^+ \mu^-$ decays is presented as an example of a direct search for New Physics. The Run-I dataset collected by the LHCb experiment, corresponding to $3$~fb$^{-1}$, is analysed by looking for an hypothetical new scalar particle produced via $B^+ \to \chi K^+$ decays and subsequently decaying into a muon pair. The signal would manifest itself as an excess in the di-muon invariant mass distribution over the Standard Model background. No significant excess is observed in the accessible ranges of mass $250 < m_\chi < 4700$~MeV$/c^2$ and lifetime $0.1 < \tau_\chi < 1000$~ps. Therefore, upper limits on the branching fraction $\mathcal{B}(B^+ \to K^+\chi(\to\mu^+ \mu^-))$ at $95\%$ confidence level are set as a function of $m_\chi$ and $\tau_\chi$. These are the most stringent limits up-to-date on the process and constrain several inflation models predicting the existence of a new light particle.