Study of beauty production and nuclear modification in p-Pb collisions through the channel $h_{\rm b} \rightarrow J/\psi + X$

Heavy-flavour hadrons, containing open or hidden charm and beauty quarks, provide one of the most important tools to test our understanding of various aspects of Quantum Chromodynamics (QCD). The heavy quark masses introduce indeed an effective threshold on the coupling of the strong interaction, allowing the hard-scattering process taking place at the initial stage of their formation to be calculated within the framework of perturbative QCD and compared to experimental measurements. In heavy-ion collisions, quarkonia and heavy-flavour hadrons play a further important role as they have long been proposed as ideal probes of the hot and deconfined medium, known as Quark-Gluon Plasma (QGP), which is thought to be produced in such collisions. The large momentum transfers involved in the formation of heavy quarks ensure their production to occur in the very first stages of the collision and allow them to traverse the medium during its full evolution. Different effects related to the presence of Cold Nuclear Matter (CNM), and not due to the formation of QGP, such as nuclear shadowing or parton energy loss, can however contribute to modify the observed yields with respect to expectations from elementary pp collisions. In this context, the study of p-Pb collisions at the LHC provides an essential tool to size the influence of CNM and to disentangle the hot and cold nuclear effects envisioned in Pb-Pb collisions. This thesis describes the production of heavy-flavour and quarkonium in heavy-ion collisions. In particular, the measurement of beauty-flavoured hadron production at mid-rapidity in p-Pb collisions at the nucleon-nucleon centre-of-mass energy $\sqrt{s_{\rm NN}}= 5.02 $ TeV is presented. The analysis has been performed on the data collected by ALICE, the LHC experiment devoted to the study of heavy-ion collisions, through the semi-inclusive decay channel $h_{\rm b} \rightarrow J/\psi + X$, where the J$\psi$ is reconstructed in the $e^{+}e^{-}$ decay channel, down to transverse momentum as low as $p_{\rm T}=1.3$ GeV/$c$. The fraction of J$\psi$ originated from beauty-hadron decays has been measured, on a statistical basis, as a function of $p_{\rm T}$ and then combined with the measurements of the inclusive J/$\psi$ production to determine the prompt J/$\psi$ cross sections. By making use of theoretical predictions based on FONLL pQCD calculations in combination with a set of nuclear parton distribution functions, the $b\bar{b}$ production has been extrapolated to derive the first measurement of the inclusive $b\bar{b}$ production cross section in p-Pb systems at LHC energies. The results have been compared to expectations derived from pp collisions at the same centre-of-mass energy in order to study possible modifications of the production due to CNM effects such as the shadowing of the gluons inside nuclei.