$CP$ phase structure of QCD from functional renormalization group

We study the low-energy properties of QCD-like theories in the presence of a $P$-odd and $U(1)$ axial breaking four-fermion operator $\left( \bar{\psi} \psi \right) \left( \bar{\psi} i \gamma_5 \psi \right)$. We apply the functional renormalization group for a low-energy effective theory involving the $CP$-violating operator. We find that allowing for the running gauge coupling, the $CP$-violating four-fermion interaction becomes relevant in the chirally broken phase. In the presence of a finite quark mass, the RG running of the $\theta$-parameter is shown to be strongly suppressed toward the infrared. The present work clarifies how strong-$CP$ effects generated at UV can non-trivially be transferred to the infrared physics in QCD-like theories.