propagators, $\delta\Omega/\delta D=0$, only the derivatives of the statistical distribution functions contribute. Using ${\text{Im}}({D_\text{T}}{\Pi_\text{T}})={\text{Re}}{D_\text{T}}{\text{Im}}{\Pi_\text{T}}+{\text{Im}}{D_\text{T}}{\text{Re}}{\Pi_\text{T}}$, the entropy density can be written as $s:=-V^{-1}\left.\partial\Omega/\partial T\right|_{\mu}=s_{g,\text{T}}+s_{g,\text{L}}+\sum_{q,s}(s_{i,+}+s_{i,-})$ with$$\begin{aligned}
s_{g,\text{T}} & = & -2d_{g}\int_{{\mathrm{d}^{4}k}}\frac{\partial{n_\text{B}}(\omega)}{\partial T}\Big\{\mbox{Im}\ln\left(+{D_\text{T}}^{-1}\right)-\mbox{Re}{D_\text{T}}\mbox{Im}{\Pi_\text{T}}\Big\},\label{eq:sgT ImLn}\\
s_{g,\text{L}} & = & -\,\,\, d_{g}\int_{{\mathrm{d}^{4}k}}\frac{\partial{n_\text{B}}(\omega)}{\partial T}\Big\{\mbox{Im}\ln\left(-{D_\text{L}}^{-1}\right)+\mbox{Re}{D_\text{L}}\mbox{Im}{\Pi_\text{L}}\Big\},\label{eq:sgL ImLn}\\
s_{q/s,\pm} & = & -2d_{q/s}\!\int_{{\mathrm{d}^{4}k}}\!\!\frac{\partial{n_\text{F}}(\omega)}{\partial T}\Big\{\mbox{Im}\ln\left(\pm S_{\pm}^{-1}\right)\mp\mbox{Re}S_{\pm}\mbox{Im}\Sigma_{\pm}\Big\},\label{eq:sq ImLn}\end{aligned}$$ each describing the entropy density of one quasiparticle species in the absence of the others. An interaction (correlation) entropy density contribution would contain terms of the form ${\text{Im}}{D_\text{T}}{\text{Re}}{\Pi_\text{T}}$ and the derivative of $\Gamma_{2}T$ with respect to the temperature. However, at 2-loop order these terms exactly cancel each other [@BIR01]. In fact, this seems to be a generic, topological feature [@CP75] which has explicitly been proven for QED [@VB98] and $\Phi^{4}$ theory [@Pes01] too.

We now focus on the terms ${\text{Im}}\ln(\pm D_{\text{T,L}}^{-1})$ and ${\text{Im}}\ln(\pm S_{\pm}^{-1})$, which can be written as$$\begin{aligned}
{\text{Im}}\ln{D_\text{T}}^{-1} & = & \arctan\left(\frac{{\text{Im}}{D_\text{T}}^{-1}}{{\text{Re}}{D_\text{T}}^{-1}}\right)+\pi\varepsilon({\text{Im}}{D_\text{T}}^{-1})\Theta\!\left(-{\text{Re}}{D_\text{T}}^{-1}\right),\label{eq:ImLn+DTm1}\\
{\text{Im}}\ln\left(-{D_\text{L}}^{-1}\right) & = & \arctan\left(\frac{{\text{Im}}{D_\text{L}}^{-1}}{{\text{Re}}{D_\text{L}}^{-1}}\right)-\pi\varepsilon({\text{Im}}{D_\text{L}}^{-1})\Theta\!\left(+{\text{Re}}{D_\text{L}}^{-1}\right).\label{eq:ImLn-DLm1}\end{aligned}$$ Similar expressions apply for the two quark propagators: one has to substitute $S_{+}^{-1}$ for ${D_\text{T}}^{-1}$ in (\[eq:ImLn+DTm1\]) and $S_{-}^{-1}$ for ${D_\text{L}}^{-1}$ in (\[eq:ImLn-DLm1\]).

From the properties of the imaginary parts of the self-energies discussed above, we find ${\varepsilon}({\text{Im}}D_{i}^{-1}(\omega))=-{\varepsilon}(\omega)$ for the gluons and ${\varepsilon}({\text{Im}}S_{\pm}(\omega))\equiv-1$ for the normal and abnormal quark branches. We end up with$$\begin{aligned}
