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If the coupling constants in QFT are promoted to functions of space-time, thedependence of the path integral on these couplings is highly constrained byconformal symmetry. We begin the present note by showing that this idea leadsto a new proof of Zamolodchikov's theorem. We then review how this simpleobservation also leads to a derivation of the a-theorem. We exemplify thegeneral procedure in some interacting theories in four space-time dimensions.We concentrate on Banks-Zaks and weakly relevant flows, which can be controlledby ordinary and conformal perturbation theories, respectively. We computeexplicitly the dependence of the path integral on the coupling constants andextract the change in the a-anomaly (this agrees with more conventionalcomputations of the same quantity). We also discuss some general properties ofthe sum rule found in arXiv:1107.3987 and study it in several examples.

The Constraints of Conformal Symmetry on RG Flows