Peccei–Quinn theory

Peccei–Quinn theory predicts that the small value of the θ parameter is explained by a dynamic field, rather than a constant value. Because particles arise within quantum fields, Peccei–Quinn theory predicts the existence of a new particle, the axion. The potential which this field carries causes it to have a value which naturally cancels, making the θ parameter uneventfully zero.

Peccei–Quinn symmetry presents θ as a functional component – a global U(1) symmetry under which a complex scalar field is charged. This symmetry is spontaneously broken by the vacuum expectation value obtained by this scalar field, and the axion is the massless Goldstone boson of this broken symmetry.

This Peccei–Quinn symmetry is inexact because it is anomalously broken by QCD instantons: If there is a compensatory term canceling the QCD anomaly breaking term, the axion becomes an exactly massless Goldstone boson and θ is no longer fixed. The effective potential of the axion is the summed 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, then the dimension 5 axion coupling term ${\displaystyle a\operatorname {Tr} [F\wedge F]}$ is suppressed by ${\displaystyle 1/\Lambda }$ where ${\displaystyle \Lambda }$ is the scale of the axion. 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 ${\displaystyle \Lambda }$ factor. This requires quite a bit of reconciliation with an approximate global symmetry, for which there is no current explanation.

• Sarkar, Utpal (2008). "Peccei–Quinn Symmetry". Particle and Astroparticle Physics. Taylor & Francis. pp. 191–197. ISBN 978-1-58488-931-1.