2d_{q/\! s}\!\int_{{\mathrm{d}^{4}k}}\!\!\left({n_\text{F}}\!+\!{n_\text{F}}^{A}\right)\!\Big\{\pi\Theta\!\left(-{\text{Re}}S_{+}^{-1}\right)-\arctan\frac{{\text{Im}}\Sigma_{+}}{{\text{Re}}S_{+}^{\text{-}1}}\,+\mbox{Re}S_{+}\mbox{Im}\Sigma_{+}\Big\},\end{aligned}$$ where the integrability condition $\partial B/\partial\Pi_{i}=\partial p/\partial\Pi_{i}$ has to be fulfilled for every quasiparticle species $i$. Thus $B$ ensures the stationarity of the thermodynamic potential under functional variation with respect to the self-energies [@GY95]. Note that the plasma frequency within the $s$-quark pressure differs from the plasma frequency within $p_{q}$ as $\mu_{s}=0$.

The pressure fully defines the model. The particle density follows by differentiation of the pressure with respect to the chemical potential at constant temperature. The Bose-Einstein distribution function ${n_\text{B}}$ does not depend on $\mu_{g}$ and strange quarks are included into the model with manifest zero net particle density, therefore $n_{g,\text{T}}=n_{g,\text{L}}=n_{s}=0$ . Due to the integrability condition, the terms containing the derivatives of the self-energies with respect to $\mu$ vanish, so that$$n\!=\! n_{q}\!\!=\!2d_{q}\!\!\int_{{\mathrm{d}^{4}k}}\!\!\left(\frac{\partial{n_\text{F}}}{\partial\mu}\!+\!\frac{\partial{n_\text{F}}^{A}}{\partial\mu}\right)\!\!\left\{ \pi\Theta\!\left(\text{-}{\text{Re}}S_{+}^{-1}\right)\!-\!\arctan\frac{{\text{Im}}\Sigma_{+}}{{\text{Re}}S_{+}^{\text{-}1}}\!+\!\mbox{Re}S_{+}\mbox{Im}\Sigma_{+}\right\} ,\label{eq:fullHTL nq}$$ thus $n_{q}(\mu\rightarrow0)\rightarrow0$.

Effective coupling
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Obviously, 2-loop QCD is only a crude approximation of the full theory. In order to accommodate non-perturbative effects in the quasiparticle model, we introduce some flexibility by parameterizing the QCD coupling constant $g^{2}$ in a phenomenologically motivated way. The truncated 2-loop running QCD coupling $g^{2}$ is given by $g^{2}(x)=16\pi^{2}(\beta_{0}\ln(x))^{-1}\,(1-2\beta_{1}\ln[\ln(x)]\,(\beta_{0}^{2}\ln(x))^{-1})$ [@PDG06], where $\beta_{0}=11N_{c}/3-2N_{f}/3$, $\beta_{1}=51-19N_{f}/3$ and $N_{f}=N_{q}+N_{s}$. It depends on the ratio $x=\bar{\mu}^{2}/\Lambda^{2}$ of the renormalization scale $\bar{\mu}$ 