# Peccei–Quinn theory

This Peccei–Quinn symmetry can't possibly be exact because it is anomalously broken by QCD instantons. If there were a compensating term canceling the QCD anomaly breaking term, the axion becomes an exactly massless Goldstone boson and θ is no longer fixed. The effective potential for the axion is the sum of the potential above the QCD scale with the potential term induced by nonperturbative QCD effects. If the axion is fundamental, or emerges at a scale far higher than the QCD scale, the dimension 5 axion coupling term $a \mathrm{Tr}[ F \wedge F ]$ is suppressed by $1/\Lambda$ where $\Lambda$ is the scale where the axion appears. Because of this, in order for θ to be so small at the minimum of the effective potential, the bare potential has to be many orders of magnitude smaller than the instanton induced potential, compounded by the $\Lambda$ factor. This requires quite a bit of adjusting for an approximate global symmetry, for which there is no current explanation.