We review recent advances in the theory of the three-dimensional dilute homogeneous Bose gas at zero and finite temperature. Effective field theory methods are used to formulate a systematic perturbative framework that can be used to calculate the properties of the system at T = 0. The perturbative expansion of these properties is essentially an expansion in the gas parameter √ na 3 , where a is the s-wave scattering length and n is the number density. In particular, the leading quantum corrections to the ground state energy density, the condensate depletion, and long-wavelength collective excitations are rederived in an efficient and economical manner. We also discuss nonuniversal effects. These effects are higher-order corrections that depend on properties of the interatomic potential other than the scattering length, such as the effective range. We critically examine various approaches to the dilute Bose gas in equilibrium at finite temperature. These include the Bogoliubov approximation, the Popov approximation, the Hartree-Fock-Bogoliubov approximation, the Φ-derivable approach, optimized perturbation theory, and renormalization group techniques. Finally, we review recent calculations of the critical temperature of the dilute Bose gas, which include 1/N -techniques, lattice simulations, self-consistent calculations, and variational perturbation theory.

We review in detail recent advances in our understanding of the phase structure and the phase transitions of hadronic matter in strong magnetic fields B and zero quark chemical potentials µ f . Many aspects of QCD are described using low-energy effective theories and models such as the MIT bag model, the hadron resonance gas model, chiral perturbation theory, the Nambu-Jona-Lasinio (NJL) model, the quark-meson (QM) model and Polyakov-loop extended versions of the NJL and QM models. We critically examine their properties and applications. This includes mean-field calculations as well as approaches beyond the mean-field approximation such as the functional renormalization group (FRG). Renormalization issues are discussed and the influence of the vacuum fluctuations on the chiral phase transition is pointed out. Magnetic catalysis at T = 0 is covered as well. We discuss recent lattice results for the thermodynamics of nonabelian gauge theories with emphasis on SU (2)c and SU (3)c. In particular, we focus on inverse magnetic catalysis around the transition temperature Tc as a competition between contributions from valence quarks and sea quarks resulting in a decrease of Tc as a function of B. Finally, we discuss recent efforts to modify models in order to reproduce the behavior observed on the lattice. A. B-dependent transition temperature T0 51 B. B-dependent coupling constant 52 XI. Anisotropic pressure and magnetization 55 XII. Conclusions and outlook 57 Acknowledgments 59 A. Notation and conventions 59 B. Sum-integrals 59 C. Small and large-B expansions 61 D. Propagators in a magnetic background 61References 63 1 Another common choice is the symmetric gauge, Aµ = 1 2 (0, By, −Bx, 0).

We calculate the free energy of a hot gluon plasma to leading order in hard-thermal-loop perturbation theory. Effects associated with screening, gluon quasiparticles, and Landau damping are resummed to all orders. The ultraviolet divergences generated by the hard-thermal-loop propagator corrections can be canceled by a counterterm which depends on the thermal gluon mass. The deviation of the hardthermal-loop free energy from lattice QCD results for T . 2T c has the correct sign and roughly the correct magnitude to be accounted for by next-to-leading order corrections. PACS numbers: 12.38.Mh, 11.10.Wx, 12.38.Cy Relativistic heavy-ion collisions will soon allow the experimental study of hadronic matter at energy densities that should exceed that required to create a quark-gluon plasma. A quantitative understanding of the properties of a quark-gluon plasma is essential in order to determine whether it has been created. Because QCD, the gauge theory that describes strong interactions, is asymptotically free, its running coupling constant a s becomes weak at sufficiently high temperatures. This would seem to make the task of understanding the high-temperature limit of hadronic matter relatively straightforward, because the problem can be attacked using perturbative methods. Unfortunately, the perturbative expansion in powers of a s does not seem to be of any quantitative use even at temperatures that are orders of magnitude higher than those achievable in heavy-ion collisions.The problem is evident in the free energy F of the quark-gluon plasma, whose weak-coupling expansion has been calculated through order a 5͞2 s [1,2]. An optimist might hope to use perturbative methods at temperatures as low as 0.3 GeV, because the running coupling constant a s ͑2pT ͒ at the scale of the lowest Matsubara frequency is about 1͞3. However, the expansion in powers of a 1͞2 s appears to converge only for extremely small values of a s . For example, if N f 6, the a 3͞2 s term is smaller than the a s term only for a s , 0.075, which corresponds to a temperature greater than 10 3 GeV. At temperatures below 1 GeV, the corrections show no sign of converging, although the convergence can be somewhat improved by using Padé approximations [3]. It is clear that a reorganization of the perturbation series is essential if perturbative calculations are to be of any quantitative use at temperatures accessible in heavy-ion collisions.The poor convergence of the perturbation series is puzzling, because lattice gauge theory calculations indicate that the free energy F of the quark-gluon plasma can be approximated by that of an ideal gas unless the temperature T is very close to the critical temperature T c for the phase transition [4,5]. The deviation of F from the free energy of an ideal gas of massless quarks and gluons is less than about 25% if T is greater than 2T c . Furthermore, the lattice results can be described surprisingly well for all T . T c by an ideal gas of quark and gluon quasiparticles with temperature-dependent masses [6].The lar...

We calculate the pressure for pure-glue QCD at high temperature to two-loop order using hard-thermal-loop (HTL) perturbation theory. At this order, all the ultraviolet divergences can be absorbed into renormalizations of the vacuum energy density and the HTL mass parameter. We determine the HTL mass parameter by a variational prescription. The resulting predictions for the pressure fail to agree with results from lattice gauge theory at temperatures for which they are available.

We calculate the three-loop thermodynamic potential of QCD at finite temperature and chemical potential(s) using the hard-thermal-loop perturbation theory (HTLpt) reorganization of finite temperature and density QCD. The resulting analytic thermodynamic potential allows us to compute the pressure, energy density, and entropy density of the quark-gluon plasma. Using these we calculate the trace anomaly, speed of sound, and second-, fourth-, and sixth-order quark number susceptibilities. For all observables considered we find good agreement between our three-loop HTLpt calculations and available lattice data for temperatures above approximately 300 MeV.