I have often read that renormalizability and scale invariance are somehow relate
ID: 1382394 • Letter: I
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.
Explanation / Answer
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.