where ${n_\text{B}}=(\exp\,(\beta\omega)-1)^{-1}$ with $\beta=1/T$ is the Bose-Einstein, and ${n_\text{F}}=(\exp(\beta(\omega-\mu))+1)^{-1}$ the Fermi-Dirac distribution function.

Application to QCD
------------------

In our present approach, the infinite sum $\Gamma_{2}$ is truncated at 2-loop order leaving the contributions exhibited e.g. in equation (25) in [@BKS07a]. Within the CJT formalism, the self-energies then follow from a functional derivative of $\Gamma_{2}$, giving the well-known 1-loop self-energies (e.g. equations (26) and (27) ibid.). In order to achieve a gauge invariant formulation of the model, we apply an additional approximation of hard thermal loops (HTL). Although originally being derived for soft external momenta $\omega,k\sim gT\ll T$, HTL results coincide with the complete one-loop results on the lightcone [@BKS07a; @Pes98b] and thus provide the correct limiting behaviour.

The resulting HTL self-energies can be found in textbooks. Here we follow the conventions of Blaizot et al. [@BIR01], where essential features of our model have been worked out, and use$$\begin{aligned}
\Pi_{\mu\nu} & = & {\Pi_\text{T}}(\omega,k)\left(\Lambda_{\text{T}}(\vec{k})\right)_{\mu\nu}-\,\,\,\,{\Pi_\text{L}}(\omega,k)\left(\Lambda_{\text{L}}(\vec{k})\right)_{\mu\nu},\\
\gamma_{0}\Sigma & = & \Sigma_{+}(\omega,k)\,\,\,\,\,\Lambda_{+}(\vec{k})\quad-\,\,\,\,\Sigma_{-}(\omega,k)\,\,\,\,\Lambda_{-}(\vec{k})\end{aligned}$$ with the scalar self-energies$$\begin{aligned}
{\Pi_\text{T}}(\omega,k)=\frac{m_{D}^{2}}{2}\left(1+\frac{\omega^{2}-k^{2}}{k^{2}}{\Pi_\text{L}}(\omega,k)\right), &  & {\Pi_\text{L}}(\omega,k)=m_{D}^{2}\left(1-\frac{\omega}{2k}\ln\frac{\omega+k}{\omega-k}\right),\label{eq:HTL PiT}\\
\text{and}\quad\quad\Sigma_{\pm}(\omega,k) & = & \frac{\hat{M}^{2}}{k}\left(1-\frac{\omega\mp k}{2k}\ln\frac{\omega+k}{\omega-k}\right),\label{eq:HTL Sigmapm}\end{aligned}$$ where $\hat{M}(T,\,\mu,\, g^{2})$ is the thermal fermion mass or plasma frequency and $m_{D}(T,\,\mu,\, g^{2})$ denotes the Debye screening mass$$\begin{aligned}
m_{D}^{2}=\!\left(\left[2N_{c}\!+\! N_{q}\!+\! N_{s}\right]T^{2}+\frac{N_{c}}{\pi^{2}}\sum_{i}\mu_{i}\right)\frac{g^{2}}{6} & \quad\text{and}\quad & \hat{M}^{2}=\frac{N_{c}^{2}-1}{16N_{c}}\left(T^{2}+\frac{\mu^{2}}{\pi^{2}}\right)g^{2}.\label{eq:mDebye 