Testing lepton flavour universality in $B^+\to K^+\pi^+\pi^-\ell^+\ell^-$ decays with LHCb

This PhD thesis aims at testing one of the fundamental properties of the Standard Model (SM) of particle physics: the lepton flavour universality (LFU), which states that $W$ and $Z$ bosons are equally coupled to the three lepton generations. In this work, processes mediated by flavour-changing neutral currents, proceeding through a $b\to s\ell\ell$ quark-level transition, are studied with great precision, probing possible deviations from LFU. Such decay modes are forbidden at tree level in the SM, and can only happen through electroweak loop or box diagrams. Hence, due to their very small SM amplitudes, these processes are sensitive to New Physics contributions that could modify their characteristics.The measurement presented in this thesis is a LFU test performed in a previously unexplored \Bu decay mode, by measuring the ratio \rkpipi between the branching fractions of \decayMM and \decayEE decays, using data collected by the \lhcb detector. The hadronic system of this decay is very rich in resonances, making these processes of particular interest to help shedding light on the spin structure of hypothetical new particles, that would explain the anomalies previously observed by the \lhcb collaboration in decays mediated by the same $b\to s\ell\ell$ transition. In addition, the measurement will provide the first observation of the \decayEE decay.The \lhcb experiment has undergone a major upgrade between 2018 and 2022, aimed at making full use of the increased \lhc luminosity. This thesis reports in particular on the new scintillating-fibre tracker, built at EPFL and other institutes, and its commissioning.