Matching Explicit and Spontaneous Scale-Invariance Breaking in Lagrangian Models

Effective Lagrangian models, constructed from fields transforming as $(3,\overline{3})+(\overline{3},3)$ and (1, 1) in S${\mathrm{U}}_3$×S${\mathrm{U}}_3$, are studied in the tree approximation to learn how solutions having broken scale and chiral symmetries are related to underlying limit solutions exhibiting spontaneous breakdown of these symmetries. The requirement of a smooth transition to the limit of scale invariance leads to restrictive conditions on the structure of the model and its solutions. For a special case of the general model, it is possible to compute explicitly the squared masses of the single-particle excitations in terms of symmetry-breaking parameters. The condition of stability ($m^2\ge 0$) leads to the allowed domains of Okubo and Mathur, and hence, provides a physical interpretation to their result. It is noted that there are several ways to normalize the ninth axial-vector current by placing its divergence and the trace of the energy-momentum tensor in a representation of ${\mathrm{U}}_1$×${\mathrm{U}}_1$.