Gravitational instantons and the quality problem of the QCD axion: Facts, speculations, and statements in between

TL;DR

This paper reexamines the impact of gravitational instantons on the QCD axion, concluding that their effects are negligible and do not threaten the axion's role in solving the strong CP problem, contrary to previous claims.

Contribution

It provides a controlled semi-classical gravity calculation showing gravitational instantons have minimal effects on the axion potential, alleviating the axion quality problem.

Findings

Gravitational instantons induce extremely small symmetry breaking effects.

The axion solution to the strong CP problem remains robust against gravitational effects.

Derived the effective axion Lagrangian including gravitational couplings.

Abstract

In this work, we critically reanalyze the explicit breaking of the Peccei-Quinn global symmetry -- and the corresponding corrections to the QCD axion potential -- induced by gravity. Specifically, we examine the role of gravitational instantons, which are non-perturbative, finite-action solutions to the Euclidean Einstein equations. These instantons represent topologically nontrivial configurations of spacetime and are analogous to instantons in gauge theory. The amount of symmetry breaking induced by gravitational instantons can be computed in a controlled way within the framework of semi-classical gravity, using 't Hooft operators, in full analogy to the computation of the axion potential arising from QCD small instanton effects. Contrary to previous results in the literature, we find that the effects of gravitational instantons are extremely small and therefore do not give rise to a significant quality problem for the axion solution to the strong CP problem, both within the Standard Model and in beyond-the-Standard-Model scenarios that involve multiple copies of the Standard Model. In conclusion, we argue that, assuming the ultraviolet completion of gravity is weakly coupled, the axion solution to the strong CP problem remains free from any quality issues due to gravity. Along the way, we derive the effective Lagrangian of the QCD axion, including its gravitational coupling.