Beautiful leptons - setting limits to New Physics with the LHCb experiment

In this thesis, searches for physics effects beyond the Standard Model are performed by studying purely leptonic and lepton-flavour violating $B$-meson decays with the LHCb experiment. Advanced statistical tools to interpret nulltests are implemented and used in the analysis of the very rare decays discussed in this thesis. A search for lepton-flavour violating $ B^+ \to K^+ \mu^±e^∓$ decays is performed based on the LHCb Run 1 sample as the first analysis of these transitions with LHCb data. Upper limits on their branching fractions are set below 10$^{-8}$, which improves previous limits by more than an order of magnitude. Similarly, $B^0_{(s)} \to e^+e^-$ decays have never before been studied with the LHCb experiment. The upper limits on their branching fractions determined in this thesis with Run 1 and part of Run 2 data improve previous limits by a factor of 30, reaching a sensitivity of $O(10^{-8})$. With this improvement the limits become sensitive to scenarios of new physics effects. In a third analysis, the $B^0_{(s)} \to \mu^+ \mu^-$ decays are studied with the full LHCb Run 1 and Run 2 data sample. The analysis is currently being reviewed by the collaboration. Its precision is expected to be similar to that of the combination of previous results of the LHC experiments obtained on a partial data set. The results obtained in this thesis impose strong constraints on popular scenarios of new physics effects.