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

TL;DR

The authors defend strong CP conservation by showing that proper treatment of the Euclidean path integral and canonical quantization requires taking the infinite-volume limit before summing over topological sectors, avoiding spurious CP-odd effects. The topological susceptibility $\chi$ is derived from the ground-state correlator in a quantum-rotor analogy and remains nonzero ($\chi=\frac{1}{4\pi^2 m}$) when limits are taken correctly, aligning with the observed $\eta'$ mass through the Witten–Veneziano mechanism. In the effective chiral theory, spurion analysis ties the physical CP phase $\xi$ to matching correlators, with the consistent choice $\xi=-\bar{\alpha}$; critiques relying on restricted EFT forms are argued to be circular. A subvolume EFT illustrates how integrating out fluctuations can produce local CP-safe descriptions, reconciling UV topological features with IR phenomenology. Overall, the work reinforces the link between topology, the $U(1)_A$ anomaly, and hadron phenomenology while clarifying how EFT parameters encode UV physics.

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 $η^\prime$-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.