an enhancement in the number of quark degrees of freedom around the critical temperature in the presence of a regulator.

The conclusion that the Pauli-Villars regularization leads to the correct asymptotics is attractive because in the NJL model with 3-dimensional cut-off the Stefan-Boltzmann limit can be reached only when the cut-off is removed in the matter integrals [@Klevansky:1994], which are finite in themselves. Such a selective removal of the cut-off must be taken “cum grano salis”, since technically the NJL model requires the presence of a finite ultraviolet cut-off throughout all integrals. Nevertheless, one might think of criticizing our result on the grounds that we seem to get the correct asymptotics due to the auxiliary and therefore unphysical Pauli-Villars terms. This is certainly not true. To see this let us return to eq. (\[J-1med\]) and consider now the contribution of the term $\sim (n_{qM}+n_{\bar qM})$ to $\nu(T)$. (Our arguments will be based on considering the simplified case $M_u^*=M_d^*=M_s^*=M_*$. We need not be rigorous in the following discussion, because the unitary symmetry breaking effects are unimportant for the asymptotics.) The integral describes the thermal energy of massive quarks, but does not contribute at large $T$ $$\label{sb-2}
     \nu_{M_*} (T) =\frac{90 N_c}{\pi^4 T^4}\int_0^\infty 