  Nambu-Jona-Lasinio-type models are often employed as low-energy models for
the theory of the strong interaction to analyze its phase structure at finite
temperature and quark chemical potential. In particular at low temperature and
large chemical potential, where the application of fully first-principles
approaches is currently difficult at best, this class of models still plays a
prominent role to guide our understanding of the dynamics of dense
strong-interaction matter. In this work, we consider a Fierz-complete version
of the Nambu-Jona-Lasinio model with two massless quark flavors and study its
renormalization group flow and fixed-point structure at leading order of the
derivative expansion of the effective action. Sum rules for the various
four-quark couplings then allow us to monitor the strength of the breaking of
the axial $U_{\rm A}(1)$ symmetry close to and above the phase boundary. We
find that the dynamics in the ten-dimensional Fierz-complete space of
four-quark couplings can only be reduced to a one-dimensional space associated
with the scalar-pseudoscalar coupling in the strict large-$N_{\rm c}$ limit.
Still, the interacting fixed point associated with this one-dimensional
subspace appears to govern the dynamics at small quark chemical potential even
beyond the large-$N_{\rm c}$ limit. At large chemical potential, corrections
beyond the large-$N_{\rm c}$ limit become important and the dynamics is
dominated by diquarks, favoring the formation of a chirally symmetric diquark
condensate. In this regime, our study suggests that the phase boundary is
shifted to higher temperatures when a Fierz-complete set of four-quark
interactions is considered.
