to the term of $\mathcal{O}(\bar q A q)$ is proportional to the tensor $$\begin{aligned}
 \tfrac{1}{2}{\cal T}^{(5)}_{\bar q A q}+{\cal T}^{(7)}_{\bar q A q}\ .\end{aligned}$$ Together with our results for the dressing functions, $z_{\bar q A
  q}^{(i)}(p,q)$ evaluated at $p^2=q^2=(p+q)^2$, Fig. \[fig:quarkgluonvertex\_components\_msp\], we conclude that this is indeed the gauge-invariant operator that determines most of the strength of the chirally symmetric non-classical tensor structures, see also Fig. \[fig:STI\]. Since the operator in contributes also to tensor structures in higher vertices, namely the two-quark-two-gluon and two-quark-three-gluon interactions, we include these as well in our truncation and dress them in accordance with gauge symmetry, see Apps. \[app:mSTIvert\] and \[trunc:qgl\] for further details. Similarly we find that the chiral symmetry breaking operator $$\begin{aligned}
\label{eq:STI_operator_II}
 \bar q\, \left(\delta_{\mu\nu} +\left[\gamma_\mu,\gamma_\nu\right]\right)  D_\mu D_\nu \,q\,,  \end{aligned}$$ contributes to $\mathcal{O}(\bar q A q)$ to the tensor $$\begin{aligned}
 \tfrac{1}{2}{\cal T}^{(2)}_{\bar q A q}+{\cal T}^{(4)}_{\bar q A q}\ .\end{aligned}$$ Since this is the most relevant operator in the phase of spontaneously broken chiral symmetry we again include the corresponding contribution to the two-quark-two-gluon vertex with gauge invariant dressing. The explicit calculation of the dressing functions of the higher interactions to check the quantitative importance of deviations from the STI which are 