Quark Matter in a Parallel Electric and Magnetic Field Background: Chiral Phase Transition and Equilibration of Chiral Density

In this article we study spontaneous chiral symmetry breaking for quark matter in the background of static and homogeneous parallel electric field $\bm E$ and magnetic field $\bm B$. We use a Nambu-Jona-Lasinio model with a local kernel interaction to compute the relevant quantities to describe chiral symmetry breaking at finite temperature for a wide range of $E$ and $B$. We study the effect of this background on inverse catalysis of chiral symmetry breaking for $E$ and $B$ of the same order of magnitude. We then focus on the effect of equilibration of chiral density, $n_5$, produced dynamically by 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, $\mu_5$, which is computed self-consistently as a function of 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 $\mu_5^2/T^2$. We find that even if chirality is produced and equilibrates within a relaxation time $\tau_M$, it does not change drastically the thermodynamics, with particular reference to the inverse catalysis induced by the external fields, as long as the average $\mu_5$ at equilibrium is not too large.