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Dilatons are pseudo-Nambu-Goldstone bosons arising from the breaking of conformal invariance. In this letter we point out that in general a dilaton mass has a power-law dependence on a small parameter related to the explicit breaking of conformal invariance whereas the ratio between the ultraviolet and infrared scales in the theory are exponentially related to the same parameter. We show that this scaling results in a separation between the dilaton mass and the infrared scale that can not be arbitrary large. Therefore a small dilaton mass necessarily is associated to a secluded conformal sector. We argue that the fact that the dilaton field must have a small displacement from the minimum of its effective potential generated near the infrared scale precludes a cosmologically interesting amount of dilatonic dark matter to be produced by a misalignment mechanism in the early Universe.
Hubisz et al. (2024) studied this question.