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Other than sharing the word “anomalous”, both the anomalous dimension in RG and the more well-known quantum anomalies (such as chiral anomaly) share a common feature. These are violations of classical symmetries at a quantum level. Anomalous dimension violates the classical scaling symmetry, and for example chiral anomaly violates the classical chiral symmetry.

My question is how correct this analogy is, and if not where does it fall apart? I know that there is a known anomaly in QFT that violates the scaling symmetry, which is the Weyl anomaly. What is its relation with the anomalous dimensions in the more mundane RG flows?

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Well, the introducing the RG scale $\mu$ breaks scaling symmetry; another way of seeing it is that the stress tensor picks up a term proportional to the beta function of the theory. At a fixed point though, scale invariance comes back, but the dimension of the fields is different from the engineering dimension. So the anomalous dimensions are there because scale invariance was broken but then came back as the renormalisation group flow ends at a fixed point, but they don't require scale invariance to be broken to exist - it only really makes sense to talk about the scaling dimension of an operator at a fixed point.

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