Infrared fixed point of the QCD vacuum angle from Berry geometry

We investigate the possibility that the physically relevant QCD vacuum angle can be interpreted as an infrared effective parameter associated with geometric structures emerging in the Yang–Mills vacuum. Working within an effective infrared framework, we consider a regime in which long-wavelength gauge configurations are treated as slow collective variables, while fermionic degrees of freedom are integrated out as fast modes. In this setting, the fermionic functional determinant may develop a geometric phase contribution analogous to a Berry holonomy, modifying the phase weighting of topological sectors in the functional integral. This motivates the introduction of an effective vacuum angle θ ₑff θ eff incorporating both the microscopic parameter and the infrared geometric response of the fermionic sector. We further propose a phenomenological evolution equation describing its scale dependence. Assuming positivity of the effective topological susceptibility, the CP-invariant configuration θ ₑff=0 θ eff = 0 emerges as a possible infrared-attractive fixed point. A simple toy model is presented to illustrate how Berry phases can induce shifts of vacuum angles in systems with topological sectors. Within this perspective, the smallness of observable strong CP violation may be interpreted as a consequence of infrared geometric effects rather than as the result of an additional symmetry or new particle degrees of freedom. The present construction is formulated as a conceptual infrared framework, rather than as a first-principles derivation within full QCD.