The discrepancies in $b\to s$ semi-leptonic decays: A complete study from model-building aspects, to the definition of novel observables, to their measurement with the LHCb detector.

Although the Standard Model (SM) of particle physics gives an overall excellent description of the observations, a few results, mainly obtained by the LHCb detector at CERN, point towards deviations in the transitions from quark $b$ to quarks $s$ and $c$. If confirmed, these anomalies would give a clear signal of physics beyond the SM, as they violate lepton flavour universality (LFU). In this context, new measurements and their theoretical predictions are crucial to define possible new physics scenarios. This thesis concerns both aspects and is therefore divided into two parts. The first one is devoted to theoretical considerations on the $b$ to $s$ transition as well as models explaining the anomalies. The second part presents a new measurement, namely the search for the decays of $B_s^0$ mesons to two muons and a photon at LHCb. Radiative leptonic decays are promising to test the SM because the additional photon not only enlarges the branching ratio by lifting the chiral suppression factor, but also offers a sensitivity to other operators. Using the language of effective field theory (EFT), the decay of the $B_s^0$ meson into two leptons and a photon is studied and new methods to reduce the theoretical uncertainty on its branching ratio are proposed. Besides, the behaviour of this decay at large dilepton mass gives the possibility of an indirect measurement where the total branching ratio is measured as a background of the corresponding non-radiative channel. Furthermore, if the violation LFU is experimentally confirmed, one also expects violation of lepton flavour. Measuring such violation, which would be an undeniable sign of new physics, is yet very challenging. Radiative decays can however support this search by offering additional channels with potentially larger branching ratios. The interpretation of the $B$ anomalies in term of shifts in the EFT coefficients put a few scenarios forward. These scenarios can then be interpreted in term of new physics models. One such model is based on the consideration of an additional symmetry group acting horizontally between the SM generations. This model is severely constrained by low energy observables, such as meson mixings and leptonic decays, but allowing for a mass degeneracy between the new group’s bosons explains all $b\to s$ anomalies while passing other experimental constraints. The absence of new physics in low energy observables can also be interpreted as the presence of leptoquarks, for which the interaction between two quark- or two lepton-currents only arises at the one loop level. A model based on a vector leptoquark can for example give an explanation to both $b\to s$ and $b\to c$ anomalies. Interestingly, ultraviolet completions of these models contain natural Dark Matter candidates, hence relating two outstanding problems of particle physics. The analysis presented in the second part is particularly challenging because the probed decay is both very rare and radiative. The main difficulty lies in the presence of a large combinatorial background due to light meson decays. Tackling it while keeping a high efficiency on the signal selection requires the use of two successive multivariate analyses. The signal is then normalised to a similar decay of the $B_s^0$ where no new physics is expected. Measuring a ratio of yields instead of an unique branching ratio allows for a partial cancellation of experimental uncertainties. On the other hand, this procedure requires a good knowledge of each selection step efficiencies for the two channels. These efficiencies are extracted from Monte-Carlo simulations or, when possible, directly from the data. Even if no significant excess is found, this analysis will allow to set the first limit on the total branching ratio.