What is the exact relationship between scale invariance and renormalizability of a theory?

Question

I have often read that renormalizability and scale invariance are somehow related. For example in this tutorial on page 12 in the first sentence of point (7), self similarity (= scale invariance ?) is referred to as the non-perturbative equivalent of renormalizability.

I don't understand what this exactly means. Can one say that all renormalizable theories are scale invariant but the converse, that every scale invariant theory is renormalizable too, is not true? I'm quite confused and I'd be happy if somebody could (in some detail) explain to me what the exact relationship between scale invariance and renormalizability is.

Answer 1

your question,

Can one say that all renormalizable theories are scale invariant but the converse, that every scale invariant theory is renormalizable too is not true?

has a sharp answer: no, one cannot say so. Renormalizable theories typically have running coupling constants with non-vanishing beta functions. The second part (what you called the 'converse') is false too. The first example that come to my mind is a theory with a spontaneously broken CFT that delivers a dilaton: the low-energy lagrangian for the dilaton is scale invariant and still is non-renormalizable having an infinite series of terms organized by the number of derivatives envolved.

The only relations I can see between scale invariance and renormalization are well known: a) renormalization typically spoils classical scale-invariance; b) a theory with strictly renormalizable terms (i.e. dimension 4 only) is classically scale invariant and it has a chance to be scale invariant at the quantum level as well; c) a non-scale invariant theory may run and approach a scale invariant theory at the end of the RG flow, either IR or UV, depending where you are heading to. This last point may be violated in very special non-unitary QFT, though.

Answer 2

There is a relation between scaling and renormalization, but not between scaling invariance and renormalization.

Scaling is possible for every polynomial Lagrangian density. In scaling, one transforms space-time, fields and coupling constants by suitable powers of a dilation factor in a way that preserves the action. Scaling is the reason for the existence of the renormalization (semi)group. It implies that the parameter defining the mass scale in a renormalization prescription is redundant.

On the other hand, scaling invariance means that the coupling constants are independent of the dilation factor, which is a rare situation. For example, QED is renormalizable and has a scaling operation affecting the electron mass, hence is not scale invariant.

Self-similarity is a property of fixed points (critical points) of a renormalization group mapping. However, in QFT this self-similarity is not a property of a physical theory but of the regularizations at different energy scales. (This energy has nothing to do with the mass scale in the renormalization prescription, though some formulas at 1 loop look very similar.)