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

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 nonperturbative, 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 semiclassical 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.