Search for flavoured leptonic decays of long-lived particles in the LHCb detector

New massive long-lived particles (LLP) are predicted by multiple beyond the Standard Model theories. This thesis presents a search of such long-lived particles decaying leptonically in the $e \mu v$ final state. The search is performed using $2.31$ $fb^{-1}$ of proton-proton collisions collected by the LHCB detector at a center-of-mass energy of $\sqrt{s}$ = 13 $TeV$, seeking for displaced vertices of electrons and muons of opposite charges. LLPs with masses between 7 and 50 $GeV/c^{2}$, and lifetimes between 2 and 50 ps, are explored in this search. Three kinds of LLP production modes are considered: the direct pair production of LLPs from quark interactions, the pair production from a Standard Model like Higgs boson with a mass of 125 $GeV/c^{2}$, and charged current production from an on-shell $W$ boson with an additional lepton. No evidence of these long-lived states has been observed, upper limits at $95 \%$ CL on the production cross-section times branching ratio have been set on the different production modes.