expected to occur below momenta of $\mathcal{O}(1$ GeV$)$ is deferred to future work.

We want to stress at this point that if we would only take into account a full basis for the quark-gluon vertex without the corresponding gauge invariant partner tensor structures in the higher vertices, we would see considerably different results. In particular, the running coupling, see Fig. \[fig:running\_couplings\], as defined from the dressing function of the classical tensor structure, $z_{\bar q A q}^{(1)}$, would deviate from the corresponding ghost-gluon running coupling at considerably larger momenta. However, the degeneracy of the running couplings defined from the different vertices at semi-perturbative momentum scales is a consequence of gauge invariance. Hence we conclude that the higher quark-gluon vertices are important for the consistency of the truncation. Moreover, since diagrams that contain the two-quark-two-gluon vertex have a different number of quark lines than the ones that contain only classical vertices, we expect a qualitative effect at finite chemical potential due to these higher quark-gluon interactions.

Note that for assessing the importance of the different tensor structures one has to take into account not only their relative strength but also their respective symmetry properties. For example, simply extracting the relative strength from 