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We show that the nonlocal two-flavor Nambu--Jona-Lasinio model predicts theenhancement of both chiral and axial symmetry breaking as the chiral imbalanceof hot QCD matter, regulated by a chiral chemical potential $\mu_5$, increases.The two crossovers are reasonably close to each other in the range of $\mu_5$examined here and the pseudocritical temperatures rise with $\mu_5$. Thecurvatures of the chiral and axial crossovers for the chiral quark chemicalpotential approximately coincide and give $\kappa_5 \simeq - 0.011$. We pointout that the presence of $\mu_5$ in thermodynamic equilibrium is inconsistentwith the fact that the chiral charge is not a Noether-conserved quantity formassive fermions. The chiral chemical potential should not, therefore, beconsidered as a true chemical potential that sets a thermodynamically stableenvironment in the massive theory, but rather than as a new coupling that mayrequire a renormalization in the ultraviolet domain. The divergence of anunrenormalized chiral density, \corr{coming from zero-point fermionicfluctuations,} is a consequence of this property. We propose a solution to thisproblem via a renormalization procedure.

Topological susceptibility, divergent chiral density and phase diagram of chirally imbalanced QCD medium at finite temperature