the sense that we also include perturbatively irrelevant operators in our truncation. Firstly, in the matter sector we include in addition the four-fermi interactions, which are required for the description of chiral symmetry breaking. Secondly, for any non-classical operator which shows a significant contribution in the flow, we identify the corresponding gauge-invariant completion and include the resulting higher order vertices in the flow equations.

The general vertex construction follows [@Fischer:2009tn]. Suppressing the explicit RG-scale dependence we parameterise $$\begin{aligned}
\nonumber 
\Gamma^{(n)}_{\Phi_1\cdots \Phi_n}(p_1,...,p_{n-1}) & \\[2ex] 
& \hspace{-1.8cm}= 
\bar \Gamma^{(n)}_{\Phi_1\cdots 
\Phi_n}(p_1,...,p_{n-1}) \prod_{i=1}^{n}\sqrt{\bar Z_{\Phi_i}(p_i)}\,, 
\label{eq:genvert}\end{aligned}$$ where we introduced a superfield $$\begin{aligned}
\label{eq:Phi}
\Phi=(A_\mu\,,\, c\,,\,\bar c\,,\,q\,,\,\bar q, \vec
\pi\,,\,\sigma\,,...)\end{aligned}$$ which subsums all dynamical degrees of freedom including the effective low-energy fields generated by dynamical hadronisation. The tensor kernel $\bar\Gamma^{(n)}_{\Phi_1\cdots\Phi_n}$ is expanded in a basis of tensor structures ${\cal T}_{\Phi_1\cdots \Phi_n}^{(i)}$ $$\begin{aligned}
\label{eq:tensorkernel} 
&\bar\Gamma^{(n)}_{\Phi_1\cdots 
\Phi_n}(p_1,...,p_{n-1})= \nonumber\\[1ex]
&\qquad \sum\limits_i z_{\Phi_1\cdots \Phi_n}^{(i)}(p_1,....,p_{n-1}) 
 {\cal T}_{\Phi_1\cdots \Phi_n}^{(i)}(p_1,...,p_{n-1}) \,.\end{aligned}$$ Since the dressing functions $z_{\Phi_1\cdots
  \Phi_n}^{(i)}(p_1,....,p_{n-1})$ depend on our choice of $\bar
Z_{\Phi_i}$, the latter are at our disposal to give special properties like RG-invariance in the perturbative regime to the former. If not specified otherwise, we choose $\bar Z_{\Phi_i}(p)\equiv
Z_{\Phi_i,k}(p)$. An important example are the classical vertices with tensor structures ${\cal 