The structure of the phase diagram for strong interactions becomes richer in the presence of a magnetic background, which enters as a new control parameter for the thermodynamics. Motivated by the relevance of this physical setting for current and future high-energy heavy-ion collision experiments and for the cosmological QCD transitions, we use the linear sigma model coupled to quarks and to Polyakov loops as an effective theory to investigate how the chiral and the deconfining transitions are affected, and present a general picture for the temperature\char21{}magnetic field phase diagram. We compute and discuss each contribution to the effective potential for the approximate order parameters, and uncover new phenomena such as the paramagnetically induced breaking of global Z3 symmetry, and possible splitting of deconfinement and chiral transitions in a strong magnetic field.

As a model for nonideal behavior in the equation of state of QCD at high density, we consider cold quark matter in perturbation theory. To second order in the strong coupling constant ${\alpha }_s,$ the results depend sensitively on the choice of the renormalization mass scale. Certain choices of this scale correspond to a strongly first order chiral transition, and generate quark stars with maximum masses and radii approximately half that of ordinary neutron stars. At the center of these stars, quarks are essentially massless.

The presence of a strong magnetic background can modify the nature and the dynamics of the chiral phase transition at finite temperature. We compute the modified effective potential in the linear sigma model with quarks to one loop in the $\overline{\mathrm{MS}}$ scheme for $N_f=2$. For fields $eB\sim 5m_{\pi }^2$ and larger a crossover is turned into a weak first-order transition. We discuss possible implications for noncentral heavy ion collisions at the Relativistic Heavy Ion Collider and the Large Hadron Collider and for the primordial QCD transition.

We study the effect of a very strong homogeneous magnetic field $B$ on the thermal deconfinement transition within the simplest phenomenological approach: the MIT bag pressure for the quark-gluon plasma and a gas of pions for the hadronic sector. Even though the model is known to be crude in numerical precision and misses the correct nature of the (crossover) transition, it provides a simple setup for the discussion of some subtleties of vacuum and thermal contributions in each phase, and should provide a reasonable qualitative description of the critical temperature in the presence of $B$. We find that the critical temperature decreases, saturating for very large fields.