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Nambu-Jona-Lasinio-type models are often employed as low-energy models forthe theory of the strong interaction to analyze its phase structure at finitetemperature and quark chemical potential. In particular at low temperature andlarge chemical potential, where the application of fully first-principlesapproaches is currently difficult at best, this class of models still plays aprominent role to guide our understanding of the dynamics of densestrong-interaction matter. In this work, we consider a Fierz-complete versionof the Nambu-Jona-Lasinio model with two massless quark flavors and study itsrenormalization group flow and fixed-point structure at leading order of thederivative expansion of the effective action. Sum rules for the variousfour-quark couplings then allow us to monitor the strength of the breaking ofthe axial $U_{\rm A}(1)$ symmetry close to and above the phase boundary. Wefind that the dynamics in the ten-dimensional Fierz-complete space offour-quark couplings can only be reduced to a one-dimensional space associatedwith the scalar-pseudoscalar coupling in the strict large-$N_{\rm c}$ limit.Still, the interacting fixed point associated with this one-dimensionalsubspace appears to govern the dynamics at small quark chemical potential evenbeyond the large-$N_{\rm c}$ limit. At large chemical potential, correctionsbeyond the large-$N_{\rm c}$ limit become important and the dynamics isdominated by diquarks, favoring the formation of a chirally symmetric diquarkcondensate. In this regime, our study suggests that the phase boundary isshifted to higher temperatures when a Fierz-complete set of four-quarkinteractions is considered.

Fierz-complete NJL model study II: towards the fixed-point and phase structure of hot and dense two-flavor QCD