s_{g,\text{T}} & = & +2d_{g}\int_{{\mathrm{d}^{4}k}}\frac{\partial{n_\text{B}}}{\partial T}\Big\{\pi\varepsilon(\omega)\Theta\!\left(-{\text{Re}}{D_\text{T}}^{-1}\right)-\arctan\frac{{\text{Im}}{\Pi_\text{T}}}{{\text{Re}}{D_\text{T}}^{\text{-}1}}+\mbox{Re}{D_\text{T}}\mbox{Im}{\Pi_\text{T}}\Big\},\label{eq:si eq theta arctan reim gT}\\
s_{g,\text{L}} & = & -\,\,\, d_{g}\int_{{\mathrm{d}^{4}k}}\frac{\partial{n_\text{B}}}{\partial T}\Big\{\pi\varepsilon(\omega)\Theta\!\left(+{\text{Re}}{D_\text{L}}^{-1}\right)-\arctan\frac{{\text{Im}}{\Pi_\text{L}}}{{\text{Re}}{D_\text{L}}^{\text{-}1}}+\mbox{Re}{D_\text{L}}\mbox{Im}{\Pi_\text{L}}\Big\},\label{eq:si eq theta arctan reim gL}\\
s_{q/s,\pm} & = & \pm2d_{q/s}\!\int_{{\mathrm{d}^{4}k}}\frac{\partial{n_\text{F}}}{\partial T}\Big\{\quad\,\,\,\pi\Theta\!\left(\mp{\text{Re}}S_{\pm}^{-1}\right)-\arctan\frac{{\text{Im}}\Sigma_{\pm}}{{\text{Re}}S_{\pm}^{\text{-}1}}\,+\mbox{Re}S_{\pm}\mbox{Im}\Sigma_{\pm}\Big\}.\label{eq:si eq theta arctan reim q}\end{aligned}$$ The partial entropy densities (\[eq:si eq theta arctan reim gT\])-(\[eq:si eq theta arctan reim q\]) and, therefore the whole entropy density expression, are independent of possible renormalization factors. As required, the expression is also explicitly UV finite, as the derivatives of the distribution functions soften the UV behavior. The terms $\pi\Theta(\ldots)$ represent the quasiparticle contributions to the entropy, while the terms containing the imaginary parts of the self-energies are related to damping effects and quasiparticle widths. In the case of HTL self-energies, Landau damping is contained within the latter terms.

The quark entropy density $s_{q/s}=s_{q/s,+}+s_{q/s,-}$ can be simplified by utilizing the parity properties for quark propagators and self-energies. Introducing the distribution function of antiparticles ${n_\text{F}}^{A}=(e^{\beta(\omega+\mu)}+1)^{-1}$ with $\partial{n_\text{F}}(-\omega)/\partial\omega=-\partial{n_\text{F}}^{A}(\omega)/\partial\omega$ and substituting $\omega\rightarrow-\omega$ within $s_{q,-}$, we find$$s_{q/s}=2d_{q/s}\int_{{\mathrm{d}^{4}k}}\left(\frac{\partial{n_\text{F}}}{\partial T}\!+\!\frac{\partial{n_\text{F}}^{A}}{\partial T}\right)\left\{ \pi\Theta\!\left(\text{-}{\text{Re}}S_{+}^{-1}\right)-\arctan\!\left(\frac{{\text{Im}}\Sigma_{+}}{{\text{Re}}S_{+}^{\text{-}1}}\!\right)+\mbox{Re}S_{+}\mbox{Im}\Sigma_{+}\right\} .\label{eq:sq combined}$$ Regarding the quasiparticle pole term $\pi\Theta(\text{-}{\text{Re}}S_{+}^{-1})$, the energy integration from $-\infty$ to $0$ gives the (anti)plasmino contribution, while the integration from $0$ to $+\infty$ delivers the contributions of the (anti)particles to the entropy density. Isolating both parts 