Heavy quark interaction in Hot QCD matter: open charm and bottom production in relativistic Heavy-Ion Collisions

The Quantum Cromodynamics(QCD) is the non-abelian gauge field theory which describes the strong interaction within the Standard Model. The basic constituents of QCD are quarks and gluons that are confined in hadronic matter: at high temperatures and/or densities quark deconfinement occurs and hadronic matter undergoes a phase transition into a new state of matter, the Quark-Gluon Plasma (QGP). Numerical solutions of QCD on lattice (lQCD) predict that such transition is properly a crossover at almost zero baryon density and with a critical temperature $T_c \approx 155$ $MeV$. The QGP state of matter is still presently studied in current Heavy-Ion Collision (HIC) experiments at the Large Hadron Collider (LHC) at CERN and at the Relativistic Heavy Ion Collider (RHIC) at the Bookhaven National Laboratory. In this scenario Heavy Quarks (HQs), mainly charm and bottom, represent the best probes to study the QGP. They are created at initial stage of HICs due to their large masses by hard perturbative QCD scattering processes and their thermalization time is comparable with the QGP lifetime. Therefore HQs can probe the entire evolution of the fireball carrying more information with respect to their light counterparts.\\ In this thesis we study the HQs dynamics within the QGP by means of a relativistic Boltzmann (BM) transport approach. In this framework we treat non-perturbative QCD effects by mean a Quasi-Particle Model (QPM) in which light quarks and gluons of the bulk are dressed with effective thermal masses and the $T$ dependence of the strength coupling is fitted to lQCD thermodynamics. In this thesis we extend our QPM-Boltzmann approach to a more realistic model in which off-shell dynamics is also taken into account. Furthermore, we discuss the dynamical evolution of charm quark elastic energy loss in a bulk medium at fixed temperature T extending the BM collision integral to include off-shell dynamics. We show the results on the transport coefficients and the time evolution of charm quark distribution function making a comparison among the Langevin dynamics, the BM collisional integral within a QPM approximation with on-shell QGP medium and the BM collision integral extended to a dynamical quasi-particles model with off-shell bulk particles. Furthermore, in this thesis we also extend our QPM approach including a momentum dependent partonic mass for the bulk particle obtained form the Dyson-Schwinger studies which yields the limit of pQCD for $p\rightarrow \infty$. A main result has been to show that the momentum dependence is an essential aspect to correctly reproduce the thermodynamics susceptibilities as recently evaluated in lattice QCD. %In this context, we also include the charm quark in our lQCD fit following the new available lattice data. A second main part of the thesis has been devoted to study for the first time new observables for the heavy-flavor sector within the full-Boltzmann transport approach and the standard QPM. In particular, we have performed realistic simulations at LHC energies for $Pb-Pb$ collisions including initial state fluctuations within a Monte-Carlo Glauber approach. The hadronization is treated by a hybrid model of coalescence plus fragmentation. We show our results on Nuclear Modification Factor $R_{AA}$ and for the elliptic flow $v_2$ and high order anisotropic flows $v_n({n\ge 2})$ of charm and bottom quarks and their hadrons. Furthermore, we extend for the first time in the heavy flavour sector this calculation in the so called Event-Shape-Engineering (ESE) technique in which the events in centrality class are divided according to magnitude of the second-order harmonic reduced flow vector $q_2$. In particular in this thesis is shown for the first time the $v_n-v_m$ correlation and the $SC(m,n)$ correlator for $D$ mesons. Furthermore, we have seen that our approach, without any parameter modification, correctly predicts the recent preliminary observables $R_{AA}$ and $v_2$ associated to bottom quark. Within this approach the extracted T-dependence of the space-diffusion coefficient $D_s$ of both charm and bottom quarks is in a agreement with lattice QCD data within the systematic uncertainties.