Broken Scale Invariance

Abstract

We study the implications of approximate scale invariance for particle masses and the dynamics of the low-lying scalar and pseudoscalar mesons. Assuming that the form factor $\theta \mathrm{}\left(t\right)\mathrm{}$ (the trace of the energy-momentum tensor) obeys an unsubtracted dispersion relation, one may derive formulas of the Goldberger-Treiman type for hadron masses if a set of scalar mesons dominates the dispersion relation. The dependence of particle masses on the coupling strength and dimension of scale-breaking operators is studied. In addition to the scale-breaking operator $u$ which breaks chiral symmetry, it seems necessary to introduce a scale-breaking $S{U}_3\times S{U}_3$ singlet $\delta$, as advocated by Wilson. The absence of $\delta$ would contradict the analysis of meson-baryon scattering lengths by von Hippel and Kim, would make the large mass of ${\eta }^{\prime }$ and the baryons difficult to understand, and would require the operator $u$ to have a dimension - 2 or mixed. At present, the evidence for $\delta$ is not completely compelling, however. In the absence of all scale breaking, we distinguish two types of theories. In the first, all masses vanish in the limit of scale invariance and the vacuum is normal under scale transformations. This is the situation envisioned in most previous research. The second alternative is that of spontaneous breakdown of scale invariance, in which case the vacuum is degenerate and most particles have masses. The latter case is especially interesting in that it requires the existence of a massless "dilaton" and further permits other masses to be not so inaccessible to extrapolations from the real world. Lagrangian models of spinless mesons are investigated to understand the interplay of scale and $S{U}_3\times S{U}_3$ transformations. Working in the limit of $S{U}_3\times S{U}_3$ symmetry, the scale breaking is due to the trilinear interactions. We consider two linear models in which there are (a) 18 meson fields belonging to $(3, \overline{3})+(\overline{3}, 3)$ and (b) 20 meson fields, with two extra mesons $S$ and $P$ belonging to (1,1) in addition to the 18 of case (a). The scale-invariant part of the Lagrangian is found to be $U_3\times U_3$ invariant, so that the trilinear coupling $\delta$ breaks ${F_0}^5$ as well as scale. We speculate that the real world is $U_3\times U_3$ invariant in the scale-invariant limit (although there exist meson models in which ${F_0}^5$ does not even exist). Spontaneous breakdown is considered for cases in which the ninth scalar (${\sigma }_0$) and the extra scalar ($S$) have nonvanishing vacuum expectation values. In the 18-meson model, spontaneous breakdown is permitted in the scale-invariant limit only when the quartic couplings have a relation prohibiting $〈{\sigma }_0〉\ne 0$ in the presence of the scale-breaking term $\delta$. The 20-meson model allows spontaneous scale breaking which survives when explicit scale breaking is introduced. In the latter model, the "dilaton" is a mixture of the $S$ and ${\sigma }_0$ mesons. The most interesting qualitative feature of the "spontaneous-breakdown" solutions is that even the canonical dimension of an operator can be somewhat ambiguous. In particular, an operator having a given dimension appropriate to a normal ground state may have different (usually several) dimensions under scale transformations appropriate to the physical ground state. DOI: http://dx.doi.org/10.1103/PhysRevD.2.2265 © 1970 The American Physical Society