Peccei–Quinn theory

In particle physics, the Peccei–Quinn theory is the best known proposal for the resolution of the strong CP problem. It was formulated by Roberto Peccei and Helen Quinn. The theory proposes that the QCD Lagrangian be extended with a CP-violating term known as the θ parameter. Because experiments have never measured a value for θ, its value must be small or zero.

Peccei–Quinn theory predicts that the θ parameter is explained by a dynamic field, rather than a constant value. Because quantum fields produce particles, 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 effectively zero.

Peccei–Quinn symmetry is a possible additional ingredient—a global U(1) symmetry under which a complex scalar field is charged. This symmetry is spontaneously broken by the VEV obtained by this scalar field, and the axion is the massless goldstone boson of this symmetry breaking. If the symmetry is a gauge symmetry then the axion is "eaten up" by the gauge boson, meaning that the gauge boson becomes massive and the axion does not exist anymore as a physical degree of freedom (see Higgs mechanism). This is phenomenologically desirable because it leaves no massless particles, which are indeed not seen experimentally.

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.