Comments on scale and conformal invariance in four dimensions

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There has been recent interest in the question of whether four dimensional scale invariant unitary quantum field theories are actually conformally invariant.

• In this note we present a complete analysis of possible scale anomalies in correlation functions of the trace of the stress-energy tensor in such theories.
• We find that 2-, 3- and 4-point functions have a non-trivial anomaly while connected higher point functions are non-anomalous.
• We pay special attention to semi-local contributions to correlators (terms with support on a set containing both coincident and separated points) and show that the anomalies in 3- and 4-point functions can be accounted for by such contributions.