Dynamical Approach to the Unitary-Symmetry Mass Formula Based on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>ω</mml:mi><mml:mo>−</mml:mo><mml:mi>ϕ</mml:mi></mml:math>Mixing

Abstract

Possible dynamical mechanisms that give rise to the unitary-symmetry mass formula of Gell-Mann and Okubo are discussed with special emphasis on a model based on $\ensuremath{\omega}\ensuremath{-}\ensuremath{\phi}$ mixing. In addition to accounting for the mass formula in a rather natural manner, the $\ensuremath{\omega}\ensuremath{-}\ensuremath{\phi}$ mixing model has the following distinctive features: (1) It explains why the mass formula fails for the vector meson octet. (2) It requires ${m}_{\ensuremath{\rho}}&lt;{m}_{M}$ provided ${m}_{\ensuremath{\pi}}&lt;{m}_{K}$. (3) It leads to $|{m}_{\ensuremath{\Xi}}\ensuremath{-}{m}_{N}|\ensuremath{\gg}|{m}_{\ensuremath{\Sigma}}\ensuremath{-}{m}_{\ensuremath{\Lambda}}|$, provided the couplings of the vector mesons to the baryons are predominantly of the $F$ type (as expected from the point of view of the conserved-vector-current theory). (4) It justifies the conjecture that it is more proper to use, in the mass formula, ${(\mathrm{mass})}^{2}$ for the mesons, but just the mass for the baryons. (5) The corrections to the mass formula are expected to be of the order of a few percent if the major contribution to the self-energies of the strongly interacting states comes from the region of a few BeV. A quantitative estimate of the $\ensuremath{\omega}\ensuremath{-}\ensuremath{\phi}$ mixing is made, and it is shown that the observed 1020-MeV $\ensuremath{\phi}$ meson (the observed 780-MeV $\ensuremath{\omega}$ meson) is about a 60-40 mixture (a 40-60 mixture) in intensity of the $T=0$ member of a unitary octet and a unitary singlet. We also show that a pair of mass formulas of the Gell-Mann-Okubo type are "self-consistent" provided the cutoff momentum (in the perturbation-theoretic sense) is much greater than a typical difference within a unitary multiplet.