Study of rare $b$-baryon decays and test of lepton universality at LHCb

Flavor-changing neutral-current $b \rightarrow s \ell^+\ell^-$ transitions are forbidden at tree level in the Standard Model (SM), and can only occur at loop level. Therefore, they are sensitive to possible New Physics (NP) effects beyond the SM. In the SM, the transitions $b \rightarrow s e^+e^-$ and $b \rightarrow s \mu^+\mu^-$ have the same probability to happen, this property is called lepton universality (LU). Probing LU in $b$-hadron decays has been recently a promising area for NP searches. Tensions with respect to the SM predictions at the level of about 2.5 standard deviations have been observed in rare $B \rightarrow K^{(*)}\ell^+\ell^-$ decays. To date, tests of LU have been performed only in decays of $b$-mesons. This thesis presents a first test of the LU in rare $b$-baryon decays $\Lambda_b^0 \rightarrow pK\ell^+\ell^-$, using the data collected by the LHCb experiment at CERN, during 2011, 2012 and 2016 data-taking periods. The expected statistical sensitivity is about 15\%, while the central value remains blind at the moment. The first measurement of the branching fraction of the decay $\Lambda_b^0 \rightarrow pK\mu^+\mu^-$ is performed with the statistical precision of 5.3\%. In addition, this thesis presents the first observation of the $\Lambda_b^0 \rightarrow pKe^+e^-$ decay, and the measurement of its branching fraction. Finally, a suppressed decay $\Xi_b^0 \rightarrow pKJ/\psi$ is observed and studied for the first time.