Strong CP as an Infrared Holonomy: The $θ$ Vacuum and Dressing in Yang-Mills Theory

TL;DR

This work reframes the strong CP problem in terms of infrared holonomy, treating the vacuum angle $\theta$ as a Berry phase of infrared-dressed states over $\mathcal{A}/\mathcal{G}$, with the Pontryagin index appearing as an integer winding. By introducing a compact collective coordinate $\phi$ and a Chern–Simons–type construction, the authors show that the holonomy $\mathcal{U}_C = \mathcal{P}\exp\left(i \oint_C \mathcal{A}_{IR}\right)$ is quantized by $Q \in \mathbb{Z}$ and yields the familiar phase $e^{i\theta Q}$. A minimal infrared model—a quantum rotor—demonstrates that many local correlators are insensitive to $\theta$ while global response functions such as the vacuum energy curvature and topological susceptibility retain $\theta$-dependence; this aligns with the Witten–Veneziano mechanism. The authors also address recent claims about $\theta$-independence tied to the order of limits, arguing that those results reflect probing a holonomy-blind subset of observables, not the absence of infrared topology. The framework offers a geometric, non-perturbative perspective on vacuum selection in Yang–Mills and QCD, with potential implications for resolving the strong CP problem through infrared representation rather than perturbative tuning.

Abstract

We reformulate the strong $CP$ problem from an infrared viewpoint in which the vacuum angle $θ$ is not treated as a local coupling but as a global Berry-type holonomy of the infrared-dressed state space over $\mathcal{A}/\mathcal{G}$. Infrared dressing is described as adiabatic parallel transport of physical states in configuration space, generated by an infrared connection $\mathcal{A}_{\rm IR}$. Using the Chern-Simons collective coordinate, we show that the Pontryagin index emerges as an integer infrared winding, such that the resulting holonomy phase is quantized by $Q\in\mathbb Z$ and reproduces the standard weight $e^{iθQ}$. A quantum rotor provides a controlled infrared example illustrating why broad classes of local correlators may remain insensitive to $θ$, while global response functions, such as the vacuum energy curvature and the topological susceptibility, retain a nontrivial dependence. We contrast this picture with recent claims of $θ$--independence based on the order of limits and show that it is consistent with both the rotor benchmark and the classic Witten-Veneziano perspective.