The Problem of Axion Quality: A Low Energy Effective Action Perspective

Any would-be Peccei-Quinn (PQ) symmetry is vulnerable to various types of explicit breaking. It has long been recognized that these can disrupt the axion solution to the strong CP problem. There have also been suggestions that, under certain circumstances, these can lead to a surprisingly large mass for the axion. Two types of corrections to this computation have been widely considered: higher dimension symmetry breaking operators, in theories where there is a complex field responsible for the spontaneous breaking of the PQ symmetry, and small instantons. Motivated by situations where small instantons dominate the $\theta$ potential of non-abelian gauge theories, we formulate the question of axion quality in terms of constraints on a Wilsonian effective action at scales somewhat above the scale of QCD. In this language, the standard axion mass computation assumes a nearly exact PQ symmetry in this action. The higher dimension operators and/or small instantons represent symmetry breaking terms in the Wilsonian action. This picture permits a uniform treatment of the problem of {\it axion quality}. If one assumes order one CP violation at high energies, then solving the strong CP problem constrains the axion potential terms in this Wilsonian action; in particular, the axion mass must be extremely close to its "standard" value.