Why dilatation generators do not generate dilatations

We show that the Ward identities associated with broken scale invariance contain anomalies in renormalized perturbation theory. In low orders, these anomalies can be absorbed into a redefinition of the scale dimensions of the fields in the theory, but in higher orders this is not possible. Also, these anomalies cannot be removed by studying the Green's functions for objects other than canonical fields, e.g., currents. These results are established to first nontrivial order in perturbation theory by explicit Feynman calculations (which give us information at all momentum transfers), and in higher orders by the method of Callan and Symanzik (which gives information only at zero momentum transfer). The two approaches are consistent within their common domain of validity. Two appendices contain self-contained treatments of the formal canonical theory of scale and conformal transformations and of the derivation of Ward identities. In another appendix, we derive the Callan-Symanzik equations for Green's functions of currents, and show that no redefinition of scale dimension is necessary for these objects, although the other anomalies remain.