and the QCD scale parameter $\Lambda$. A term involving $\ln^{-2}(\bar{\mu}^{2}/\Lambda^{2})$ which is only a small correction for $\bar{\mu}^{2}\approx\Lambda^{2}$ was neglected.

The renormalization scale is usually taken to be the first Matsubara frequency $2\pi T$, while the latter one is just a parameter to be adjusted using experimental data. Introducing the pseudocritical temperature of QCD matter at vanishing net baryon density $T_{c}$ and substituting $\Lambda\rightarrow2\pi T_{c}/\lambda$, the ratio $\bar{\mu}/\Lambda$ becomes $\lambda T/T_{c}$. In order to avoid the Landau pole of $g^{2}(T/T_{c})$ at $T_{c}$ a temperature shift with parameter $T_{s}$ is introduced, replacing $x$ by $\xi=\lambda(T-T_{s})/T_{c}$. The result $$G^{2}(T\geq T_{c},\mu=0)=\frac{16\pi^{2}}{\beta_{0}\ln\xi^{2}}\left(1-\frac{2\beta_{1}}{\beta_{0}^{2}}\frac{\ln\left[\ln\xi^{2}\right]}{\ln\xi^{2}}\right)\label{eq:eff coupling}$$ is our *effective coupling*. Within the plasma phase for temperatures $T>T_{c}/\lambda+T_{s}$ it is well-behaved; however, at some point within the hadronic phase, i.e. below $T_{c}$, an infrared (IR) divergence does occur. In order to prevent this divergence a phenomenological infrared cutoff for $G^{2}$ can be applied. See [@Blu04a] for details.

Adjustment to lattice calculations
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The two QPM parameters $\lambda$ and $T_{s}$ have to be adjusted to results of numerical first-principle QCD calculations dubbed lattice data. Most of the past work on the QPM has been tested against lattice data from [@KLP00] for rather large and temperature dependent lattice restmasses of 