Scalar Mesons, Hadron Masses, and Approximate Scale Invariance

Abstract

The possible role of a scalar-meson nonet in determining hadron masses by means of generalized Goldberger-Treiman formulas is studied. Existing data on scalar-pseudoscalar meson couplings are analyzed, assuming ${\mathrm{SU}}_3$-invariant couplings for the unmixed mesons. The $\frac{d}{f}$ ratio for the coupling of octet scalars to the baryon octet is predicted to be the same as in the Gell-Mann-Okubo mass formula. The strength of the transition vacuum → scalar meson (induced by the energy-momentum tensor ${\theta }_{\mu \nu }$) is measured by the constant $F_{\sigma }$. Denoting the ${\mathrm{SU}}_3$ singlet scalar meson by ${\sigma }_0$ and the octet $\eta$-like scalar by ${\sigma }_8$, we find $F_{{\sigma }_0}$ to be much larger than $F_{{\sigma }_8}$, provided that the ${\sigma }_0\mathrm{BB}$ coupling is not considerably smaller than the ${\sigma }_8\mathrm{BB}$ coupling. Then $F_{{\sigma }_0}$ is comparable to the usual pion decay constant $F_{\pi }$. The pseudoscalar octet dispersion relation is not scalar dominated. This result is also suggested by analysis of partial conservation of axial-vector current for the scalar-pseudoscalar system, and consideration of the relation to the underlying scale-invariant limit. Implications for the underlying dynamics are discussed.