Clearing up the Strong $CP$ problem

TL;DR

This paper revisits solutions to the Strong CP problem, showing gauged discrete symmetries can preserve CP and arguing the problem remains a genuine issue in QCD, with implications for model building.

Contribution

It demonstrates that gauged discrete symmetries can preserve CP, countering recent claims, and defends the reality of the Strong CP problem against recent arguments.

Findings

Gauged discrete symmetries can preserve CP in QCD.

The Strong CP problem remains valid, with nonzero neutron EDM arising from QCD dynamics.

Gauged discrete models face model-building challenges but are not fundamentally obstructed.

Abstract

The absence of a neutron electric dipole moment (EDM) constrains the quantum chromodynamics (QCD) theta angle to be less than one part in ten billion, posing the Strong $CP$ problem. We revisit two classes of proposed solutions. First, we show that when $P$ or $CP$ is realized as a gauged discrete symmetry - as can arise in quantum gravity - the vacuum necessarily preserves $CP$, contrary to recent claims that discrete-symmetry solutions fail. Gauged discrete models face model-building challenges, such as avoiding contributions to the neutron EDM after spontaneous $P$ or $CP$ breaking, but in principle have no fundamental obstructions. Second, we critically examine recent arguments that the Strong $CP$ problem is illusory, demonstrating that a nonzero neutron EDM at finite $\bar\theta$ follows directly from well-understood QCD dynamics. Taken together, our results reinforce the reality of the Strong $CP$ problem and highlight gauged discrete-symmetry realizations of $P$ or $CP$ as plausible solutions.