Indirect searches for new physics beyond the Standard Model rely on comparing the properties of various processes to their Standard Model predictions. Leptonic and semileptonic decays of heavy-quark hadrons offer an attractive laboratory for indirect searches for potential new particles or interactions. The leptonic decay modes of $B$ mesons, such as ${B^+ \to \mu^+ \nu_\mu}$, are challenging to study in hadron-collider experiments owing to their single-charged-track topology. To tackle this challenge, the emission of a virtual photon, decaying into a dilepton, allows to turn the one-track topology into a three-track one, allowing to determine the $B$-meson decay vertex position. This leads to a much simpler experimental measurement at hadron colliders. Since these processes have a neutrino in the final state, it is beneficial to validate our experimental understanding of the virtual photon emission, using fully reconstructed decay modes of $B$ mesons.

This thesis presents the first study of one such process: ${B^+ \to K^+ J/\psi \mu^+\mu^-}$, which is formed by an emission of a virtual photon from the abundant tree-level decay ${B^+ \to J/\psi K^+}$. It is shown that, beyond the virtual-photon contribution, other processes need to be taken into account, that come from light vector mesons decaying to dimuons, but also excited charmonium states decaying to ${J/\psi \mu\mu}$. The total rate of the ${B^+ \to K^+ J/\psi \mu^+\mu^-}$ process, including resonant contributions, is measured, using the data collected by the LHCb experiment in 2016--2018. It is compatible, within uncertainty, with the expectations, but a larger dataset is needed to achieve a firm conclusion.

This measurement serves as a proof-of-concept for the rich multilepton physics programme that will become accessible with the large dataset to be collected by upgraded LHCb detector in the coming years (Run 3 and after).