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

We show that the nonlocal two-flavor Nambu--Jona-Lasinio model predicts the enhancement of both chiral and axial symmetry breaking as the chiral imbalance of hot QCD matter, regulated by a chiral chemical potential $μ_5$, increases. The two crossovers are reasonably close to each other in the range of $μ_5$ examined here and the pseudocritical temperatures rise with $μ_5$. The curvatures of the chiral and axial crossovers for the chiral quark chemical potential approximately coincide and give $κ_5 \simeq - 0.011$. We point out that the presence of $μ_5$ in thermodynamic equilibrium is inconsistent with the fact that the chiral charge is not a Noether-conserved quantity for massive fermions. The chiral chemical potential should not, therefore, be considered as a true chemical potential that sets a thermodynamically stable environment in the massive theory, but rather than as a new coupling that may require a renormalization in the ultraviolet domain. The divergence of an unrenormalized chiral density, \corr{coming from zero-point fermionic fluctuations,} is a consequence of this property. We propose a solution to this problem via a renormalization procedure.