section \[sec:fullHTL\] and investigated in some detail. Finally, a conclusion is given in section \[sec:conclusion\].

Derivation of the full HTL model\[sec:derivation\]
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The effective action
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A connection of the fundamental theory of QCD and the thermodynamic potential of the QGP is provided by the Luttinger-Ward formalism [@LW60; @Bay62] as shown in [@BKS07a]. Alternatively, the Cornwall-Jackiw-Tomboulis (CJT) formalism [@CJT74] may be used, as a translationally invariant QGP in equilibrium and without spontaneously broken symmetries is considered. In this case, both formalisms are equivalent [@Ris03].

The CJT formalism requires the stationarity of the effective action$$\begin{aligned}
\Gamma[D,S]=I & -\frac{1}{2}\left\{ \text{Tr}\left[\ln D^{-1}\right]+\text{Tr}\left[D_{0}^{-1}D-1\right]\right\} \nonumber \\
 & +\quad\left\{ \text{Tr}\left[\ln S^{-1}\right]+\text{Tr}\left[S_{0}^{-1}S-1\right]\right\} \,\,+\,\,\Gamma_{2}[D,S],\label{eq:cjt effective action}\end{aligned}$$ where $I$ is the classical action and $D$ and $S$ are the full gluon and quark propagators while the subscript $0$ denotes the respective free equivalents. The functional $\Gamma_{2}$ represents the sum over all two-particle irreducible skeleton graphs of the theory. The traces $\text{Tr}$ contain the integration over the four-dimensional phase space as well as a trace $\text{tr}$ over discrete indices. The integration is performed using the imaginary time formalism [@LeB96; @Kap89; @YHM95]. For the grand canonical potential $\Omega=-T\Gamma$ [@Bro92; @Riv88] this yields$$\begin{aligned}
\frac{\Omega}{V} & = & \mbox{tr}\!\!\int\!\!\frac{{\mathrm{d}^{4}k}}{(2\pi)^{4}}{n_\text{B}}(\omega)\,\mbox{Im}\!\left(\ln D^{-1}-\Pi D\right)+2\,\mbox{tr}\!\!\int\!\!\frac{{\mathrm{d}^{4}k}}{(2\pi)^{4}}{n_\text{F}}(\omega)\,\mbox{Im}\!\left(\ln S^{-1}-\Sigma S\right)-\frac{T}{V}\Gamma_{2},\label{eq:Omegafinal}\end{aligned}$$ 