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A particle on a ring or: how I learned to stop worrying and love $θ$-vacua
Mohammad Aghaie, Ryosuke Sato·January 26, 2026
hep-phhep-thQuantum Physics
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Abstract
Recently, Ai, Cruz, Garbrecht, and Tamarit (ACGT)~\cite{Ai:2020ptm, Ai:2024vfa, Ai:2024cnp, Ai:2025quf} claimed that there is no strong CP problem by adopting a new order of limits in the volume and topological sector. We critically examine this proposal by focusing on simple one-dimensional quantum mechanics on a ring. We demonstrate that consistent results are obtained only when one sums over all topological sectors \textit{before} taking the large $T$ limit. This observation justifies the conventional path integral formulation of gauge theories and implies that the strong CP problem does exist in QCD.