It is often stated that points in the space of quantum field theories for which all parameters are invariant under renormalisation – that is to say, fixed points of the RG evolution – are scale-invariant field theories. Certainly this should be a necessary condition, since the theory must look the same at all scales. However, I have doubts whether this is sufficient.
In the classical theory, there is no notion of renormalisation, but nevertheless we can talk about scale-invariant theories; these theories possess a global dilatation symmetry. Theories with inherent mass scales, such as $\phi^4$ theory with a non-zero quadratic term, are not classically conformal because they lack such a symmetry. It seems to me that the vanishing (or perhaps blowing up?) of such dimensionful couplings must also be a condition to impose on a quantum field theory, if we wish for it to be scale-invariant.
My question is then: are the dimensionful couplings of all fixed points in RG flow necessarily trivial? Is it impossible for an RG fixed point to have some mass scale $M \neq 0$ in its Lagrangian?
As I see it, either the answer is yes, in which case being at a fixed point is enough to guarantee scale invariance, or the answer is no, in which case we also need to make an additional requirement of our theories if we wish for them to be conformal, beyond the vanishing of all beta functions