Artificial first-order phase transition in a magnetized Nambu--Jona-Lasinio model with a quark anomalous magnetic moment

Recently, first-order phase transitions have been predicted as an effect of the inclusion of quark anomalous magnetic moment (AMM) in the hot and magnetized Nambu--Jona-Lasinio model (NJL). These transitions appear in the chiral condensate for different combinations of AMM and magnetic fields and could lead to inverse magnetic catalysis. However, in this work, we show that the predicted first-order phase transitions are related to regularization-dependent issues. To show this, we explore, in the context of the vacuum magnetic regularization (VMR) scheme, two different scenarios: when mass-dependent (MD) and mass-independent (MI) terms are present in the subtraction of the divergences. In the MD case, as we increase the AMM value, it is observed the appearance of a nonmassive minimum in the thermodynamical potential, which induces a first-order phase transition from the massive minimum. We argue that the MD terms must be avoided in order to satisfy the predictions of Lattice QCD, and we propose a MI solution that is valid in the limit which the magnetic fields are smaller than the squared of vacuum effective quark mass.