On the magnetic catalysis and inverse catalysis of phase transitions in the linear sigma model

We consider the evolution of critical temperature both for the formation of a pion charged condensate as well as for the chiral transition, from the perspective of the linear sigma model, in the background of a magnetic field. We developed the discussion for the pion condensate in one loop approximation for the effective potential getting magnetic catalysis for high values of B, i.e a raising of the critical temperatur e with the magnetic field. For the analysis of the chiral restoration, we go beyond this approximation, by taking one loop thermo-magnetic corrections to the couplings as well as plasma screening effects for the boson’s masses, expressed through the resumati on of ring diagrams. Here we found the opposite behavior, i.e. inverse magnetic catalysis, i.e. a decreasing of the chiral critical temperature a s function of the intensity of the magnetic field, which seems t o be in agreement with recent results form the lattice community.