to be a much more compelling argument for fixing $C(T,\mu )$ than the assumption made that the different cut-off procedures must give the same result when cut-offs are removed.

A clear insight into the origin of this result can be obtained by considering the contribution of $C(T,\mu )$ and the term $\sim (n_{q0}+n_{\bar q0})$ in eq. (\[J-1med\]) to the number of effective degrees of freedom $\nu (T)$, because exactly these terms determine the correct low-temperature behavior of the function $\nu (T)$. We designate this contribution by $\nu_\Lambda (T)$, and plot it in fig. \[fig-1a\]. Thus, we consider the following function of temperature $$\label{intconst} 
     \nu_\Lambda (T)=\frac{90 N_c}{\pi^4 T^4}\int_0^\infty |\vec{p}_E|^4
     {\mathrm{d}}|\vec{p}_E|\,(1-\hat\rho_{\Lambda \vec{p}_E})
     \frac{n_{q0}+n_{\bar q0}}{E_p(0)}$$ calculated at a fixed value of the cut-off parameter $\Lambda$ (the value taken is the same for both parameter sets considered, see Table I). At first glance one might wish to associate $\nu_\Lambda (T)$ with the contribution of massless states, introduced to $J_{-1}^{\mathrm{med}}$ by assigning the low limit $M=0$ to the integral (\[J-1\]). However, this expectation is potentially fallacious. One can easily see from eq. (\[intconst\]) that $\nu_{\Lambda}(T)$ vanishes for all $T$ if the the integral is not regularized $\hat\rho_{\Lambda \vec{p}_E}\to 1$. Hence only the auxiliary Pauli-Villars states 