Infrared Dressing and the Strong CP Problem: Geometric Renormalization of the Vacuum Angle

We revisit the strong CP problem from the viewpoint of the infrared structure of non-Abelian gauge theories. In Yang-Mills theory, motion between topologically inequivalent vacua may be described in terms of a compact collective coordinate associated with the Chern-Simons number. Implementing an adiabatic separation between slow topological modes and fast gluonic fluctuations leads to a reduced Born-Oppenheimer Hamiltonian governing the infrared dynamics. We show that the physical parameter entering this reduced Hamiltonian is not the bare vacuum angle $θ$, but an effective holonomy $θ_{\rm eff}$ that includes a Berry phase induced by the fast gluonic sector. The induced holonomy becomes a self-consistent response function of the infrared dressing, leading to a nonperturbative renormalization group flow for $θ_{\rm eff}$. This infrared flow admits CP-invariant fixed points toward which the effective vacuum angle is dynamically driven in the infrared limit. In this framework, CP violation is not forbidden by the fundamental theory but becomes dynamically suppressed along the infrared flow generated by adiabatic dressing. The strong CP problem is thus realized as a nonperturbative infrared relaxation mechanism governed by the Berry response of the fast gluonic sector, without the introduction of additional dynamical fields.