Consistency Conditions for Chiral Symmetry and Scale Invariance. I. Dimensions of Operators

It is demonstrated that in the contemporary models of chiral-symmetry and scale-invariance breaking, the matrix elements of the trace of the energy-momentum tensor $\Theta$ are necessarily rapidly varying in the low-energy region. It is further shown that the limit of exact scale invariance cannot be achieved smoothly. An alternative energy-momentum tensor is proposed, which is related to the "new improved" tensor with finite matrix elements in perturbation theory, and whose matrix elements do not possess this singular behavior. By demanding maximal smoothness, the scale dimension of the divergence of the axial-vector current is found to be one.