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An axion can be understood as an oscillation of the axion field in some potential.

In the Higgs mechanism, there are 4 degrees of freedom. The Higgs boson is the radial oscillation and 3 remaining degrees of freedom are Nambu-Goldstone bosons which are eaten by the gauge bosons.

In the Peccei-Quinn mechanism we only have 2 degrees of freedom. When the universe has a temperature above the $T_{QCD}$, before the vacuum realignment mechanism (VEM) kicks in, we have a symmetrical mexican hat potential. The axions are Nambu-Goldstone bosons "oscillating" along the azimuthal angle inside the brim, and eventually become Pseudo-Nambu-Goldstone bosons after the VEM is turned on.

My question is what particle corresponds to radial motion in this potential?

Vacuum Realignment Mechanism

Taken from this paper

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This particle doesn't have a standard name -- if pressed, most people would probably just say "the radial part of the Peccei-Quinn field".

The reason for this is twofold. First, you generically expect it to be quite heavy, on the order of the Peccei-Quinn scale, and hence far above what can be probed in experiments. Second, its particular properties depend very sensitively on how you set up the model. For example, if you had two coupled PQ-charged complex scalars and broke $U(1)_{\text{PQ}}$, you'd still get one light axion, but now you'd have three extra heavy particles rather than one. So the really interesting part of the model is that which is simultaneously generic and observable, in other words, the axion itself.

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