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

TL;DR

The paper critically evaluates a recent proposal to solve the strong CP problem by changing the order of limits in the Euclidean path integral, using exactly solvable quantum mechanical models, and finds it fails to reproduce correct spectra.

Contribution

It provides a critical analysis of the ACGT proposal using quantum mechanical models, demonstrating the failure of their limit prescription to match physical spectra.

Findings

ACGT procedure does not reproduce the correct energy spectrum

The spectrum is a key physical observable that the procedure fails to match

Conclusions about CP conservation based on the proposal are unjustified

Abstract

Recently, Ai, Cruz, Garbrecht, and Tamarit (arXiv:2001.07152, arXiv:2404.16026, arXiv:2511.04216) claimed that the strong CP problem can be avoided by adopting a particular order of limits in the Euclidean path integral, in which the spacetime volume is taken to infinity before summing over all topological sectors. We critically examine this proposal using exactly solvable examples of one-dimensional quantum mechanics on a ring, namely the quantum rotor and the quantum pendulum. These systems provide fully controlled settings with known $\theta$-dependent spectra. We find that the ACGT procedure fails to reproduce the correct energy spectrum. Since the spectrum is a direct physical observable, this result demonstrates that the proposed order of limits cannot be justified and conclusions about CP conservation in QCD cannot be based on this prescription alone.