Scale Invariance and Spectral Forms for Conformal Stress Tensors

For theories with dimensionless coupling constants, it is shown in the context of source theory that unsubtracted single spectral forms for gravitational (stress tensor) vertices occur whenever the tensor which is invariant in the scaling limit is used to provide the primitive coupling of particles to gravity. Ward identities corresponding to gauge and scale invariance are derived, and it is observed that the former is ambiguous unless the requirement of consistency with scaling be imposed. Several examples of the theorem and identities are presented; in particular it is found that the dilation Ward identity fails for the Yukawa interaction, but this failure cannot be associated with an anomalous dimension. In this case, there is no anomaly in the vertex function, for the requirement of gauge invariance suffices to fix all subtractions. This pleasing state of affairs holds only when one of the basis tensors coincides with the conformal stress tensor.