Conformal symmetry and cluster decomposition?

Question

I would expect a conformal field theory would not satisfy a cluster decomposition of correlation functions. This may be due to my lack of understanding of conformal symmetry, but I would think a theory which is scale invariant would have interactions which do not get weaker at long distances, and so connected correlators would remain as large as the unconnected correlators.

I know that conformal field theories do typically satisfy cluster decomposition, so what about my thought process is flawed?

Answer 1

Correlations do get weaker at long distances in unitary CFT. The unitarity bound on the scaling dimension of operators forbids the existence of correlation functions that are constant or get stronger with the distance.

And then there is the operator product expansion that tells you that if you have a bunch of operator is some region of space-time A that is space-like separated from another bunch of operators in a region B, then you can approximate this with a single operator in region A and a single operator in region B. This is approximately a 2-point correlation function, and it decreases with distance.

(What differs in CFT compared with QFT with a mass gap is that the correlation between regions A and B decreases like a power of the distance, and not exponentially fast.)