Perspectives on QCD, Topology and the Strong CP Problem

TL;DR

This paper offers a careful critique of the strong CP problem by separating what follows strictly from local gauge invariance and causality from what requires global topological structure. It analyzes how the θ-term interacts with topology, the definition of topological charge, and nonperturbative effects like instantons, while highlighting the role of the axial anomaly in relating quark-mass phases to the vacuum angle. The authors show that observable CP violation in QCD depends on the invariant combination {bar}{θ} = θ + arg det M} and explain how a massless quark would render strong CP violation unphysical, thereby clarifying the conceptual underpinnings of the problem. Importantly, the work preserves the standard motivations for axions and axion-like particles, outlining that their existence remains a compelling extension of the Standard Model notwithstanding the clarified locality-based perspective. Overall, the article sharpens the foundational assumptions behind the strong CP problem and emphasizes the distinction between local dynamics and global topological input in QCD.

Abstract

On the basis of allowed local gauge symmetries, the QCD Lagrangian admits a CP-violating term proportional to the topological charge density, commonly referred to as the $θ$ term. A priori, any value of $θ$ is consistent with the local symmetries of the theory, while current experimental limits constrain $θ\lesssim 10^{-10}$. The apparent extreme smallness of this parameter is known as the strong $CP$ problem. In this work, we provide a careful critical overview of the conceptual assumptions underlying the $θ$ term, focusing on the roles of topology, the definition of topological charge density, rough gauge field configurations, instantons, and anomalies. We contrast the assumptions required to describe QCD at nonzero $θ$ with those sufficient at $θ= 0$, and argue that a vanishing $θ$ term is compatible with a formulation based solely on local gauge invariance and causal locality, without invoking additional global structure. The perspective developed here is intended as a conceptual analysis of the standard formulation of the strong $CP$ problem. It does not challenge the internal consistency of QCD at nonzero $θ$, nor does it diminish the independent theoretical and phenomenological motivation for axion and axion-like particle physics, which are well-motivated extensions of the Standard Model.