Study of J/$\psi$ production with the LHCb experiment in proton-proton collisions at $\sqrt{s} = 7$ TeV

The LHCb detector is one of the four main experiments installed at the proton-proton Large Hadron Collider (LHC) at CERN, Geneva. The first beams circulated in the collider in September 2008 and the data taking started in April 2010. The LHCb detector is a single-arm forward spectrometer conceived to extensively study the CP violation in the $B$ meson system, looking for constraints of the Standard Model predictions, for possible effects of new physics beyond this theory and generally for rare phenomena in the $b$ and $c$ quark sectors with high precision. Already in the first data recorded by the experiment, a huge amount of J/$\psi$ mesons have been collected, which allow to make an extensive study of the charmonia production. This is a very interesting sector in particle physics. From the experimental point of view, the measurement of the charmonium production cross section and polarization has been carried on by several experiments. At Tevatron, CDF measured the $J/\psi$ and $\psi (2S)$ cross section and the polarization with Run I and Run II data. HERA-B at DESY performed a measurement of the $J/\psi$ polarization with proton collisions on fixed target. Also the PHENIX collaboration at RHIC recently published a measurement of the $J/\psi$ polarization with $pp$ collisions. The experimental results obtained for the $J/\psi$ polarization are hardly comparable among each other because performed in different kinematic ranges and reference frames. For what concerns the theory predictions many models have been proposed in the recent years but none of them is able to fit at the same time the charmonium production cross section and polarization. The present work aims at giving a preliminary measurement of the prompt $J/\psi$ polarization, binned in $J/\psi$ transverse momentum and rapidity, and of the double differential production cross section (in transverse momentum and rapidity bins) of the $J/\psi$ prompt component: this is done analysing the $J/\psi$ decays into a muon pair. The cross section is calculated taking into account the measured polarization. Moreover the double differential production cross section, in transverse momentum and rapidity, of the $J/\psi$ component coming from the $b$-hadrons decays is presented. From this last result the $b\overline{b}$ production cross section is extrapolated from the LHCb angular coverage to the $4 \pi$ solid angle. The discussion is organized in eight chapters. The first chapter contains an introduction to the physical motivations that drive the interests in the charmonium production and a description of the most recent theoretical models: a presentation of the physical observables and a panoramic view of the experimental situation is also given. In the second chapter an overall description of the LHCb detector is given. The experimental conditions of the 2010 data taking are also described. The third chapter will give some basic introduction to the track reconstruction and the muon identification procedures in LHCb. The fourth chapter will describe the selection of the interesting signal events and the procedure to disentangle the prompt and from $b$ $J/\psi$ components. In the fifth and sixth chapter the polarization and cross section measurements are described and the obtained results are presented. In the last chapter the conclusions and a summary of the analysis will be drawn.