Constraints on RG Flows from Protected Operators

We consider protected operators with the same conformal dimensions in the ultraviolet and infrared fixed point. We derive a sum rule for the difference between the two-point function coefficient of these operators in the ultraviolet and infrared fixed point which depends on the two-point function of the scalar operator. In even dimensional conformal field theories, scalar operators with exactly integer conformal dimensions are associated with Type-B conformal anomalies. The sum rule, in these cases, computes differences between Type-B anomaly coefficients. We argue the positivity of this difference in cases in which the conformal manifold contains weakly coupled theories. The results are tested in free theories as well as in $\mathcal N = 2$ superconformal QCD, necklace quivers and holographic RG flows. We further derive sum rules for currents and stress tensor two-point functions.