Scale invariance and dilaton mass

We consider a generic scale invariant scalar quantum field theory and its symmetry breakdown. Based on the dimension counting identity, we give a concise proof that dilaton is exactly massless at the classical level if scale invariance is broken spontaneously. On the other hand, on the basis of the generalized dimension counting identity, we prove that the dilaton becomes massive at the quantum level if scale invariance is explicitly broken by quantum anomaly. It is pointed out that a subtlety occurs when scale invariance is spontaneously broken through a scale invariant regularization method where the renormalization scale is replaced with the dilaton field. In this case, the dilaton remains massless even at the quantum level after spontaneous symmetry breakdown of scale symmetry, but when the massless dilaton couples non-minimally to the Einstein-Hilbert term and is applied for cosmology, it is phenomenologically ruled out by solar system tests unless its coupling to matters is much suppressed compared to the gravitational interaction.