from a full basis to avoid ambiguities in the projection procedure. Furthermore, note that in the purely fermionic theory all ten channels diverge at the chiral symmetry breaking scale signaling resonant quark-anti-quark states, as illustrated in Fig. \[fig:4fermirebos\] for the $(\sigma-\pi)-$channel. These divergencies are a consequence of ignoring momentum dependencies and can be removed by dynamically hadronising only the $(\sigma-\pi)$-channel. Nevertheless it would be interesting to also bosonise other four-fermi channels to investigate the properties of the corresponding bound states. Alternatively an investigation of the momentum dependencies of the four-fermi interactions themselves is also conceivable. As a word of caution, our statements about the relative strength of four-fermi channels are only valid in the vacuum as in particular finite chemical potential is expected to shift the relative strength of four-fermi interaction channels.

Finally we want to mention that Fig. \[fig:fourfermis\_msp\] captures only the zero external momentum limit of the four-fermi interaction channels. Although it was necessary to calculate the momentum dependence of the $s$-channel momentum configuration for the evaluation of the quark-propagator for this work, we postpone a thorough discussion of the momentum dependence of the four-fermi interactions to future publications. Here we only note that the effect of such 