Properties of hot and dense strongly interacting matter

In this thesis we consider effective models of quantum chromodynamics to learn about the chiral- and deconfinement phase transitions. In Chapter 1 we review basic properties of strongly interacting matter and the foundations of finite temperature field theory. We review furthermore the nonperturbative functional renormalization group (FRG) approach. In Chapter 2 we introduce the quark-meson (QM) model and its extensions including the Polyakov-loop variables and repulsive vector interactions between quarks. We then discuss features of the model both in the mean-field approximation and in the renormalization group treatment. A novel method to solve the renormalization group equations based on the Chebyshev polynomials is presented at the end of the chapter. In Chapter 3 the scaling behavior of the order parameter at the chiral phase transition is studied within effective models. We explore universal and nonuniversal structures near the critical point. These include the scaling functions, the leading corrections to scaling and the corresponding size of the scaling window as well as their dependence on an external symmetry breaking field. We consider two models in the mean-field approximation, the QM and the Polyakov-loop-extended quark-meson (PQM) models, and compare their critical properties with a purely bosonic theory, the O(N) linear sigma model in the N → ∞ limit. In these models the order parameter scaling function is found analytically using the high temperature expansion of the thermodynamic potential. The effects of a gluonic background on the nonuniversal scaling parameters are studied within the PQM model. Furthermore, numerical calculations of the scaling function and the scaling window are performed in the QM model using the FRG. Chapter 4 contains a study of the critical properties of net-baryon-number fluctuations at the chiral restoration transition in a medium at finite temperature and net baryon density. The chiral dynamics of quantum chromodynamics is modeled by the PQM Lagrangian, that includes the coupling of quarks to vector meson and temporal gauge fields. The FRG is employed to properly account for the O(4) criticality at the phase boundary. We focus on the properties and systematics of ratios of the net-baryon-number cumulants near the phase boundary. The results are presented in the context of the recent experimental data of the STAR Collaboration on fluctuations of the net proton number in heavy-ion collisions at RHIC. We show that the model results for the energy dependence of the cumulant ratios are in good overall agreement with the data, with one exception. At center-of-mass energies below 19.6 GeV, we find that the measured fourth-order cumulant deviates considerably from the model results, which incorporate the expected O(4) and Z(2) criticality. We assess the influence of model assumptions and in particular of repulsive vector-interactions, which are used to modify the location of the critical endpoint in the model, on the cumulants ratios. Then finally in Chapter 5 we explore the influence of finite-volume effects on baryon-number fluctuations in the quark-meson model. We employ the FRG in a finite volume, using a smooth regulator function in momentum space. We compare the results for a smooth regulator with those for a sharp (or Litim) regulator, and show that in a finite volume, the latter may produce unwanted artifacts. In a finite volume there are only apparent critical points, about which we compute the ratio of the fourth- to the second-order cumulant of quark number fluctuations. When the volume is sufficiently small the system has two apparent critical points; as the system size decreases, the location of the apparent critical point can move to higher temperature and lower chemical potential. At the end of the thesis, conventions are collected in Appendix A. Basic properties of the unitary groups are presented in Appendix B. In Appendix C, definition and relations of Chebyshev polynomials are provided. Finally, in Appendix D initial conditions and model parameters are given