Search for the $B^0_d\to\mu^+\mu^-$ decay and measurement of the $B^0_s\to\mu^+\mu^-$ branching fraction and effective lifetime

The search for new physics is the current aim of particle physics and might be pursued $\textit{directly}$, by producing new possible particles in high energy collisions, or $\textit{indirectly}$, by measurements of processes in which loops of new virtual particles might affect, for example, the decay rate. Being not limited by the collision energy, indirect searches are sensitive to particle masses which are larger than those accessible in direct searches. For this reason, indirect searches are a powerful tool to probe heavy particles that cannot be produced at colliders. The $B^0_{d,s}\to\mu^+\mu^-$ decays are among the most sensitive probes to physics beyond the Standard Model. Such decays are extremely rare, occurring few times in billions of $B$ decays, due to loop and helicity suppressions. The decay probability is however precisely predicted in the Standard Model, as the purely leptonic final state allows to condensate hadronic interactions into a single constant. The search for $B^0_{d,s}\to\mu^+\mu^-$ decays, started more than 30 years ago, finally came into success at LHC with the LHCb and CMS joint observation of the $B^0_s\to\mu^+\mu^-$ process, at a rate in agreement with the Standard Model expectation. The precision achieved in the $B^0_s\to\mu^+\mu^-$ branching fraction measurement enormously constrained New Physics models. However, the search continues: there is still room for New Physics effects in the $B^0_s\to\mu^+\mu^-$ mode but new observables and precision measurements are required. In addition, a possible observation of the $B^0_d\to\mu^+\mu^-$ decay at a rate above its prediction, still allowed by the present experimental constraints, would be an unambiguous sign of New Physics. In particular, the current experimental results suggest a slight enhancement of the $B^0_d\to\mu^+\mu^-$ rate, which strongly pushed for a step further in the analysis, concerning both statistics and background rejection power. In this thesis, the LHCb measurement using Run 1 and a fraction of Run 2 data is presented. This work represents a substantial step towards the aforementioned goal, since the rejection and the estimate of the backgrounds have been greatly improved. Thanks to the enlarged statistics, together with a new and optimised analysis, the first single-experiment observation of the $B^0_s\to\mu^+\mu^-$ decay, and the first measurement of its effective lifetime, were made possible.