Spontaneously broken scale invariance and minimal fields of canonical dimensionality

We address the question whether a scale invariant theory can contain interacting minimal fields of canonical dimensionality. It is known that the answer to this question is negative provided the scale symmetry is respected by the ground state. We present arguments showing that generally it is not the case if the scale symmetry is broken spontaneously. An example of a theory containing interacting scalar fields of dimension one is given. An apparent contradiction between these two statements arises in the high energy limit when it seems that the symmetry breaking phase can be approximated by the symmetric one. We provide two different scenarios to reconcile the contradiction.