New massive long-lived particles (LLP) are predicted by multiple beyond the Standard Model theories. This thesis presents the search of such long-lived particles decaying leptonically in the $e \mu \nu$ 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 to $50 $ GeV/c$^2$, and lifetime 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 mode.