Chiral Symmetry Restoration with a Chiral Chemical Potential: the Role of Momentum Dependent Quark Self-energy

In this article we study restoration of chiral symmetry at finite temperature for quark matter with a chiral chemical potential, $\mu_5$, by means of a nonlocal Nambu-Jona-Lasinio model. This model allows to introduce in the simplest way possible a Euclidean momentum, $p_E$, dependent quark mass function which decays (neglecting logarithms) as $1/p_E^2$ for large $p_E$, in agreement with asymptotic behaviour expected in QCD in presence of a nonperturbative quark condensate. We focus on the critical temperature for chiral symmetry restoration in the chiral limit, $T_c$, versus $\mu_5$, as well as on the order of the phase transition. We find that $T_c$ increases with $\mu_5$, and that the transition remains of the second order for the whole range of $\mu_5$ considered.