it is advantageous to describe these interactions in terms of composite operators which is achieved via the introduction of scale-dependent mesonic field operators [@Gies:2001nw; @Pawlowski:2005xe; @Floerchinger:2009uf; @Braun:2014ata]. In the present work we follow the dynamical hadronisation procedure set-up in [@Pawlowski:2005xe; @Braun:2014ata]. In each renormalisation group step this leads e.g. to $\lambda_{\pi}\rightarrow h_\pi^2/(2
m_{\pi}^2)$ at vanishing $s$-channel momentum in the four-fermi channel. Here $\lambda_{\pi}$ corresponds to the exchange of pions with mass $m_\pi$ and Yukawa coupling $h_\pi$. This exact dynamical change of field-variables avoids the numerically inconvenient singularity shown in Fig. \[fig:4fermirebos\], which is a consequence of neglected momentum dependencies in the four-fermi interaction. Additionally, it provides a smooth transition from QCD degrees of freedom to constituent quarks and light mesons as low-energy effective degrees of freedom. The resulting low-energy description in terms of a quark-meson model introduces no model parameter dependence provided the UV initial scale is chosen large enough $\Lambda_\text{UV}\gg\Lambda_\text{QCD}$, see Sec. \[sec:4fermi\] for an explicit demonstration. Finally we want to stress that the restriction to such a small set of low-energy degrees of freedom is justified by the comparably large masses in the remainder of the spectrum of the strong interaction. Since any of the hadrons can play 