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The common argument for vanishing Goldstone boson is like, if there is some massless particle generated from the spontaneous symmetry breaking, it should be detected. Since we never saw that, it is not possible.

However, if there is some particle which has not been found, it could be (a) very heavy; (b) very weakly interacting. The Goldstone particle was rejected based on reason (a). Could (b) somehow be true? e.g. as a candidate of dark matter? Or since the Higgs mechanism was confirmed, this possibility (very weakly interacting) is already ruled out?

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A Goldstone boson is a generic type of particle formed when symmetries are spontaneously broken. If you want to suggest that dark matter is a Goldstone boson then that says very little unless you suggest a specific model with a symmetry to be broken.

When exact symmetries are broken you get a massless Goldstone boson (except in a few special circustances, E.g. in Gauge theory the extra mode gives mass to the gauge bosons instead of forming a Goldstone boson) Dark matter cannot be formed from massless particles since they would not be gravitationally bound to galaxies and we know that dark matter is. Massless particles would fly past on the same trajectories as photons in the microwave background.

If the broken symmetry is not perfect you get pseudo-Golstone bosons which are light on the scale of the model, but not massless. The pion is an example from flavour chiral symmetry breaking, but it is not stable. Any theory that predicted such a particle would predict other new particles that could just as easily be part of dark matter if they are stable. Without a specific proposal for such a theory not much has been said.

Note that it is actually very easy to dream up particle models of dark matter, e.g. you just need a new quantum number to explain stability. The difficulty is to find a theory that is well motivated from other considerations. e.g supersymmetry solves the hierarchy problem etc., axions solve the strong CP problem. However there is no clear reason why dark matter needs to solve other problems in this way. Until we can detect a signature for dark matter interactions it is going to be very hard to settle what it is.

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    $\begingroup$ Incidentally, Steven Weinberg recently suggested that Goldstones of a dark U(1) have left traces in the CMB along with the neutrinos, arxiv.org/abs/1305.1971 $\endgroup$ Nov 3 '13 at 18:23
  • $\begingroup$ Could you explain the phrase "which are light on the scale of the model" ? For example, pions are light w.r.t what scale? @PhilipGibbs-inactive $\endgroup$
    – SRS
    Oct 31 '19 at 14:50

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