Functional renormalization group study of a four-fermion model with $CP$ violation

We perform a functional renormalization group analysis of a four-fermion model with $CP$ and $P$ violation to explore the nonperturbative infrared dynamics of quantum chromodynamics (QCD) within the Wilsonian renormalization group framework, particularly in the context of spontaneous $CP$-violation models. Our analysis of the fixed-point structure reveals that, in the large-$N_c$ limit, the $CP$-violating $\bar{\theta}$ parameter is dynamically induced and approaches $\pi \cdot (N_f/2)$ (where $N_f$ is the number of flavors) as the system enters the chirally broken phase. This behavior arises due to criticality and the large anomalous dimensions of the $U(1)_A$-violating four-fermion couplings. Furthermore, this trend appears to persist beyond the leading large-$N_c$ approximation, provided that the infrared dynamics of QCD remains dominated by the scalar condensate of the quark bilinear, as expected. Notably, our findings highlight that $CP$-violating four-fermion interactions, which are perturbatively irrelevant, can become relevant in the chirally broken phase through nonperturbative effects, with potential implications for spontaneous $CP$-violation scenarios.