sufficient to strongly suppress the $\eta'$-channel in comparison to the resonant pion channel. Additional contributions due to the $U(1)_A$-anomaly would lead to an even stronger suppression, see [@Pawlowski:1996ch; @Mitter:2013fxa]. Additionally, for sufficiently large initial scales, these anomalous contributions are suppressed relative to the contribution that orginates from the explicit symmetry breaking due to non-vanishing current quark masses. First checks indeed indicate that the anomalous contributions at a sufficiently initial large cutoff scale do not play a quantitatively important rôle, a more detailed study will be presented elsewhere. On the other hand, anomalous contributions corresponding to fluctuations below the cutoff scale are already taken into account by integrating the FRG running. This has also been demonstrated e.g. for the quantum mechanical anharmonic oscillator [@Zappala:2001nv]. Therefore, all but the resonant pion four-fermi channel constitute subleading contributions with a quantitative effect of less than $5$ %, see Fig. \[fig:4fermirebos\]. Independent of their relative strength, the suppression of any of the four-fermi interactions is overcome by the strength of $\alpha_s$ only in the non-perturbative regime of QCD at scales of $\mathcal{O}(1$ GeV$)$.

In the light of these results it is sufficient to take into account only the $(\sigma-\pi)$-channel, provided one uses a projection obtained 