Reply to "Clearing up the Strong $CP$ problem"

TL;DR

This paper defends the conservation of $CP$ symmetry in QCD against recent critiques, clarifying the role of limits and effective theories, and reaffirming the validity of previous results showing $CP$ invariance.

Contribution

It refutes recent criticisms of the $CP$ conservation proof in QCD, clarifies the role of the order of limits, and emphasizes the importance of general coupling choices in effective theories.

Findings

The topological susceptibility is fixed by canonical commutation relations, independent of limit order.

Critique based on low-energy effective theory assumptions is invalid due to restricted coupling choices.

Previous results on $CP$ conservation in QCD are reaffirmed as consistent and correct.

Abstract

The conservation of $CP$ in QCD has been shown to follow from a careful treatment of the path integral and canonical quantization in arXiv:2001.07152 and arXiv:2403.00747. Here, we refute the critique of these results put forth in arXiv:2510.18951. First, using the quantum rotor as an analogue of QCD, it is argued in arXiv:2510.18951 that the topological susceptibility vanishes when using the limiting procedure of arXiv:2001.07152. When translated to QCD, this would contradict the observed $\eta^\prime$-mass. We show that this is not the case because the susceptibility is defined from the vacuum correlator of the topological charge density, which for the rotor is just fixed by the canonical commutation relation. The latter does not depend on the disputed order of limits. Second, it is suggested in arXiv:2510.18951 that $CP$ violation in QCD can be established by considering the low-energy effective theory alone. We show that here the argument relies on assuming from the start choices of couplings that lead to $CP$ violation but are not of the most general form allowed by spurion analysis. No valid reason is given for why allowed choices leading to $CP$ conservation, that match the computation of ultraviolet correlators as shown in arXiv:2001.07152 and arXiv:2403.00747, would be inconsistent.