In Euclidean space $C={\gamma }_2{\gamma }_0$ satisfies the usual relations: $C=-C^{-1}=-C^T=-C^{†}$ and $C{\gamma }_{\mu }^T{C}^{-1}=-{\gamma }_{\mu }$.

In the non-Abelian case $(D_{\mu }{j}_{\mu }{)}^a=0$ instead of ${\partial }_{\mu }{j}_{\mu }^a=0$. On the other hand, the conserved color current associated with global color symmetry is given by $J_{\mu }^a=j_{\mu }^a+(j_{\mathrm{g},\mathrm{g}\mathrm{h}}{)}_{\mu }^a$, where $j_{\mathrm{g},\mathrm{g}\mathrm{h}}$ contains gluons and ghosts.

The point here is that the regularized expression has divergent and finite terms. Gauge-variant artifacts, which are hidden in the finite terms, must be eliminated by counterterms, which however contain divergent and finite pieces as well as gauge-variant pieces if the regularization is gauge variant.

To avoid confusion, we emphasize that ${\text{tr}}_{D,G}[{\mathcal{S}}^D{\gamma }_{\nu }]={\text{tr}}_D[({\mathcal{S}}_{11}^D+{\mathcal{S}}_{22}^D){\gamma }_{\nu }]$ is not the quark number current, which in the Nambu-Gor’kov bases is instead ${\text{tr}}_D[({\mathcal{S}}_{11}^D-{\mathcal{S}}_{22}^D){\gamma }_{\nu }]$, and is nonzero for $\nu =0$.

Even in normal quark matter, this contribution should be taken into account because the mass in the QCD vacuum, the effective mass $M_{\chi }$, differs from the current mass in chirally restored normal quark matter. Usual hard-dense-loop calculations tacitly avoid this gauge-variant artifact by using the current quark mass $m$ in the chirally symmetric vacuum.

The remaining part does not contain the interaction so that it can yield only a cut instead of poles.

If we wish to find the vertex correction, ${\delta }_{\mathrm{v}}{\mathrm{\Pi }}_{\mu \nu }$, one can—instead of calculating it explicitly—compute $C_{\text{gaps}}$ and ${\mathrm{\Pi }}_{\mu \nu }$ for the bare vertex, and then use them to read off ${\delta }_{\mathrm{v}}{\mathrm{\Pi }}_{\mu \nu }$. In particular, when we consider the damping of gap functions in the UV, we can set $C_{\text{gaps}}=0$ and directly relate ${\mathrm{\Pi }}_{\mu \nu }^L$ to $-{\delta }_{\mathrm{v}}{\mathrm{\Pi }}_{\mu \nu }$ because ${\mathrm{\Pi }}_{\mu \nu }^L+{\delta }_{\mathrm{v}}{\mathrm{\Pi }}_{\mu \nu }=0$.

The behavior of the quark-gluon vertex in vacuum can be quite different for different choices of gauge-fixing conditions. This discussion is beyond our scope in this work.

This description may be misleading because a number of studies have indicated that the confined and Higgs phases can be smoothly connected [30, 31]. Indications are based on gauge-invariant correlation functions in which quarks and gluons are not separately discussed and all excitations are composite, with no color charges. Such a connection between the two phases is obscured in gauge-fixed computations using quark and gluon propagators, such as that performed here.