T}^{(5)}_{\bar q A q}\,,
  \qquad {\cal T}^{(6)}_{\bar q A q}\,,
  \qquad{\cal T}^{(7)}_{\bar q A q}\,,\end{aligned}$$ and four tensors which break chiral symmetry $$\begin{aligned}
  {\cal T}^{(2)}_{\bar q A q}\,,
  \qquad{\cal T}^{(3)}_{\bar q A q}\,,
  \qquad{\cal T}^{(4)}_{\bar q A q}\,,
    \qquad{\cal T}^{(8)}_{\bar q A q}\,,  \end{aligned}$$ listed explicitly in App. \[trunc:qgl\]. Each of the eight tensor structures leads to a contribution in the effective action that, if separated from the remainder of the action, violates gauge invariance. For example, the classical tensor structure, $\gamma^\mu$, corresponds to the term $\bar q
\slashed A q$ in the effective action which is by itself not gauge invariant. However, it appears as part of the gauge-covariant derivative $\bar q \slashed D q$ which respects gauge invariance. On the other hand, for the additional tensor structures, ${\cal
  T}^{(i)}_{\bar q A q}$, $i>1$, such a gauge-covariant completion is not automatically included. A naïve inclusion of these tensor structures alone would therefore violate gauge invariance in the form of (modified) Slavnov-Taylor identities, see the discussion in App. \[app:mSTI\].

In the semi-perturbative, chirally symmetric regime we find the gauge-invariant operator $$\begin{aligned}
\label{eq:STI_operator_I}
 {\text{i}}\sqrt{ 4 \pi \alpha_{s} }  \, \bar q\, \gamma_5
  \gamma_\mu \,\epsilon_{\mu\nu\rho\sigma} \{ F_{\nu\rho}\,,\,
  D_\sigma\} \,q\,,  \end{aligned}$$ whose contribution 