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I study various properties of the critical limits of correlators containinginsertions of conserved and anomalous currents. In particular, I show that theimprovement term of the stress tensor can be fixed unambiguously, studying theRG interpolation between the UV and IR limits. The removal of the improvementambiguity is encoded in a variational principle, which makes use of sum rulesfor the trace anomalies a and a'. Compatible results follow from the analysisof the RG equations. I perform a number of self-consistency checks and discussthe issues in a large set of theories.Comment: 15 page

A note on the improvement ambiguity of the stress tensor and the critical limits of correlation functions