contribution of the massive Pauli-Villars states, will not affect the leading term as well.

One might worry that the low-temperature expansion of the integral (\[limit\]) starts from the unphysical contribution which corresponds to the massless quark states. This fear would be valid if the potential had not the term $C(T,\mu )$. Let us fix the $M_f$-independent function $C(T,\mu )$ in eq. (\[effpot-t\]) to avoid the problem. For that it is instructive to compare the matter quark-loop part of the thermodynamic potential obtained here (i.e. the $J^{\mathrm{med}}_{-1}$-part) with the corresponding result of the standard NJL approach. There is only one difference between such calculations: we use the Pauli-Villars subtractions instead of a 3-dimensional cut-off $\Lambda_3$. Therefore we can expect that if one removes the regularizations in both approaches ($\hat{\rho}_{\Lambda\vec{p}_E}\to 1$ and $\Lambda_3\to\infty$) the finite matter part of the thermodynamic potentials must coincide. From this requirement of consistency we find $$\label{FTP}
       C(T,\mu )= -\frac{N_c}{\pi^2}\int_0^\infty{\mathrm{d}}|\vec{p}_E||\vec{p}_E|^4
       \frac{n_{q0}+n_{\bar q0}}{E_p(0)}\ \to\ C(T,0)
       =-\frac{7N_cN_f}{180}\pi^2T^4.$$ As a result, the unwanted massless quark contribution to the vacuum energy at small $T$ disappears.

Effective number of quark degrees of freedom
--------------------------------------------

A quantity of interest related with the thermodynamic potential is the number of quark degrees of freedom present at 