Are fixed points of RG evolution really scale-invariant?

Question

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

Answer 1

No, dimensionful couplings do not have to be all set to zero at an RG fixed point. An RG fixed point is one where all of the beta functions vanish, and beta functions generally have the form $$\beta(g_i) = (d_i - d) g_i + \hbar A_{ij} g_j + \ldots$$ where $d_i$ is the dimension of the corresponding operator. If one truncates the series at $O(\hbar^0)$ then the only possible solution is to have $g_i = 0$ if $d_i \neq 0$, so in classical field theory the only fixed points are the massless free theory and massless $\phi^4$ theory.

In a quantum field theory we must account for loop diagrams, which give terms that are higher order in $\hbar$. Then the zeroes of the beta functions are completely different; the massless free theory remains a fixed point, called the Gaussian fixed point, but massless $\phi^4$ theory acquires a mass scale by dimensional transmutation. But this process can also work in reverse. In this case there's a new fixed point, the Wilson-Fisher fixed point, where the classical mass term is nonzero. This is dimensional transmutation running in reverse; the mass renormalizes to exactly zero.

Answer 2

If all the couplings are zero then you are sitting on the trivial Gaussian fixed point. Being a fixed point is characterized by the vanishing of the beta functions (some derivative of the couplings as functions of scale), not that of the couplings themselves.

Also, in general, scale invariance does not imply conformal invariance. You need something extra like a traceless energy-momentum tensor. The study of this issue is an active area of research, see the review "Scale invariance vs conformal invariance" by Nakayama.