Phase diagram of QCD in a magnetic field

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 mu(f) are reviewed in detail. Many aspects of QCD are described using low-energy effective theories and models such as the bag model, the hadron resonance gas model, chiral perturbation theory (chi PT), the Nambu-Jona-Lasinio (NJL) model, the quark-meson (QM) model, and Polyakov-loop extended versions of the NJL and QM models. Their properties and applications are critically examined. This includes mean-field calculations as well as approaches beyond the mean-field approximation such as the functional renormalization group. Renormalization issues are discussed and the influence of the vacuum fluctuations on the chiral phase transition is pointed out. At T = 0, model calculations and lattice simulations predict magnetic catalysis: The quark condensate increases as a function of the magnetic field. This is covered in detail. Recent lattice results for the thermodynamics of non-Abelian gauge theories with emphasis on SU(2)(c) and SU(3)(c) are also discussed. In particular, inverse magnetic catalysis around the transition temperature T-c as a competition between contributions from valence quarks and sea quarks resulting in a decrease of T-c as a function of B is focused on. Finally, recent efforts to modify models in order to reproduce the behavior observed on the lattice are discussed.