Quark matter in a parallel electric and magnetic field background: Chiral phase transition and equilibration of chiral density

Abstract

In this article, we study spontaneous chiral symmetry breaking for quark matter in the background of static and homogeneous parallel electric field $\mathbit{E}$ and magnetic field $\mathbit{B}$. We use a Nambu-Jona-Lasinio model with a local kernel interaction to compute the relevant quantities to describe chiral symmetry breaking at a finite temperature for a wide range of $E$ and $B$. We study the effect of this background on the inverse catalysis of chiral symmetry breaking for $E$ and $B$ of the same order of magnitude. We then focus on the effect of the equilibration of chiral density ${n}_{5}$, produced dynamically by an axial anomaly on the critical temperature. The equilibration of ${n}_{5}$, a consequence of chirality-flipping processes in the thermal bath, allows for the introduction of the chiral chemical potential ${\ensuremath{\mu}}_{5}$, which is computed self-consistently as a function of the temperature and field strength by coupling the number equation to the gap equation and solving the two within an expansion in $E/{T}^{2}$, $B/{T}^{2}$, and ${\ensuremath{\mu}}_{5}^{2}/{T}^{2}$. We find that even if chirality is produced and equilibrates within a relaxation time ${\ensuremath{\tau}}_{M}$, it does not change drastically the thermodynamics, with particular reference to the inverse catalysis induced by the external fields, as long as the average ${\ensuremath{\mu}}_{5}$ at equilibrium is not too large.