Critical Endpoint and Inverse Magnetic Catalysis for Finite Temperature and Density Quark Matter in a Magnetic Background

In this article we study chiral symmetry breaking for quark matter in a magnetic background, $bm B$, at finite temperature and quark chemical potential, $mu$, making use of the Ginzburg-Landau effective action formalism. As a microscopic model to compute the effective action we use the renormalized quark-meson model. Our main goal is to study the evolution of the critical endpoint, ${cal CP}$, as a function of the magnetic field strength, and investigate on the realization of inverse magnetic catalysis at finite chemical potential. We find that the phase transition at zero chemical potential is always of the second order; for small and intermediate values of $bm B$, ${cal CP}$ moves towards small $mu$, while for larger $bm B$ it moves towards moderately larger values of $mu$. Our results are in agreement with the inverse magnetic catalysis scenario at finite chemical potential and not too large values of the magnetic field, while at larger $bm B$ direct magnetic catalysis sets in.