Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A particle on a ring or: how I learned to stop worrying and love $θ$-vacua

Mohammad Aghaie, Ryosuke Sato

TL;DR

The paper tests the claim that a special order of limits eliminates θ-dependence in gauge theories by analyzing a quantum rotor and a quantum pendulum on a ring. It demonstrates that consistent, θ-dependent observables arise only when summing over all topological sectors before taking the large-time limit, aligning canonical and path-integral results. The topological susceptibility is shown to be a genuine global effect, χ = 1/(4π^2 I) for the rotor and analogous results for the pendulum, which contradicts the ACGT prescription. The findings uphold the conventional understanding of θ-vacua in QCD and highlight the importance of global topological sums in defining observables, with implications for the strong CP problem.

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.

A particle on a ring or: how I learned to stop worrying and love $θ$-vacua

TL;DR

The paper tests the claim that a special order of limits eliminates θ-dependence in gauge theories by analyzing a quantum rotor and a quantum pendulum on a ring. It demonstrates that consistent, θ-dependent observables arise only when summing over all topological sectors before taking the large-time limit, aligning canonical and path-integral results. The topological susceptibility is shown to be a genuine global effect, χ = 1/(4π^2 I) for the rotor and analogous results for the pendulum, which contradicts the ACGT prescription. The findings uphold the conventional understanding of θ-vacua in QCD and highlight the importance of global topological sums in defining observables, with implications for the strong CP problem.

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 limit. This observation justifies the conventional path integral formulation of gauge theories and implies that the strong CP problem does exist in QCD.
Paper Structure (13 sections, 109 equations)

This paper contains 13 sections, 109 equations.

Table of Contents

  1. Introduction
  2. A ring without potential: quantum rotor
  3. Canonical quantization and energy spectrum
  4. Propagator
  5. The partition function and energy spectra
  6. Topological susceptibility
  7. A ring with potential: quantum pendulum
  8. The partition function
  9. The partition function and topological susceptibility by ACGT method
  10. Order of Limits, Topological Sectors, and theta-vacua
  11. Conclusion
  12. Aharonov-Bohm effect for a charged particle on a ring
  13. Functional determinant formula for KB