Scale-Invariance and Chiral-Symmetry Breaking for Nucleon States

Abstract

Assuming that chiral invariance is only broken by terms in the energy-momentum tensor ${\Theta }^{00}$ belonging to the $(3,\overline{3})+(\overline{3},3)$ representation of $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$, we derive a sum rule for the matrix element of $\Theta$ (the trace of the energy-momentum tensor) between nucleon and pion-nucleon states which may suggest that the divergence of the axial-vector current has dimension 1. We then go on to consider an analogous sum rule which suggests that the energy-momentum tensor contains a scale-invariance-breaking chiral singlet term, and finally we point out some implications for $\mathrm{SU}\left(3\right)\mathrm{}$ breaking of the form of matrix elements of $\Theta$.