Perspectives on QCD, Topology and the Strong CP Problem

Abstract

The strong $CP$ problem occupies a unique position in modern particle physics, connecting nonperturbative quantum chromodynamics, topology, anomalies and the global structure of gauge theories. It has motivated extensive theoretical developments, including instantons, the $\theta$ vacuum, lattice studies of topological charge, and the Peccei-Quinn mechanism and its associated axion phenomenology. At the same time, the relationship between local quantum field theory, global topology and the conventional construction of the $\theta$ vacuum continues to raise important conceptual and mathematical questions. This review presents a pedagogical overview of the strong $CP$ problem from the perspectives of its historical development, physical motivation and mathematical formulation. Beginning with the conventional formulation of the problem and the local structure of QCD, we review the roles of the axial anomaly, topological charge density, topological susceptibility, instantons, lattice gauge theory, and the Hamiltonian and Euclidean formulations of nonperturbative QCD. A recurring theme is the distinction between local structures that follow directly from relativistic quantum field theory and additional global constructions commonly employed in the conventional treatment of topological sectors and $\theta$ vacua. The review clarifies which aspects of the conventional framework are direct consequences of established local quantum field theory and which rely on additional mathematical assumptions concerning global topology. In particular, standard nonperturbative results involving local correlation functions and topological susceptibility can be formulated without requiring the full global construction of topological sectors and the $\theta$ vacuum. Alternative viewpoints and outstanding conceptual questions are also discussed.

96 pages, 3 figures