Two-flavor chirally imbalanced quark matter beyond large-$N_c$

We investigate a chirally imbalanced medium in the context of the two-flavor Nambu--Jona-Lasinio model using both the large-$N_c$ (LN) and beyond large-$N_c$ (BLN) approximations. To incorporate BLN effects, we consider the optimized perturbation theory (OPT) to the first nontrivial order, which includes two-loop (exchange) contributions. This procedure allows us to explicitly explore how finite $N_c$ corrections affect the thermodynamics as well as the phase diagram of chirally imbalanced quark matter. We then compare the results obtained with a sharp three-dimensional cutoff -- generically referred to as the traditional regularization scheme (TRS) -- and with an alternative procedure called the medium separation scheme (MSS). In the first case, we observe that the pseudocritical temperature decreases as the chiral chemical potential increases, an effect dubbed inverse chiral catalysis (ICC). On the other hand, when considering the MSS regularization, which properly isolates the medium contributions from the vacuum, we find the opposite result. We show that the results obtained with MSS are consistent with well-established LQCD data in both the LN and BLN approximations. Finally, we suggest that to cope with the high-density limit, the standard OPT interpolation prescription must be modified with the inclusion of an extra variational parameter.