Analysis of $\rho$ condensation in a Nambu-Jona-Lasinio model

A non-local Nambu-Jona-Lasinio model can be derived from QCD in the low-energy limit. In this way, it is possible to fix all the free parameters of the model with physical ones. We used this approach to derive a local limit to the Nambu-Jona-Lasinio model, with the parameters those obtained from QCD, in order to fix the physical parameters of $\rho$ condensation and obtain the critical magnetic field. $\rho$ condensation is a consequence of the highly non-trivial behavior of the QCD vacuum in presence of a very strong magnetic field giving rising to superconductive behavior in quark matter. We extend previous computations by including finite temperature terms.