Phase structure of a Z(2) scalar-fermion model

General broken-SU(3) sum rules for baryon transitions B′→(12)±+P (B′ denotes a nonet or octet baryon with arbitrary spin and parity) are derived. The approach is based on the use of a chiral SU(3)⊗SU(3) charge algebra, the hypothesis of partially conserved axial-vector current (PCAC) and, in particular, the assumption of asymptotic SU(3) symmetry formulated for the matrix elements of the vector charge VK. The VK is the SU(3) raising or lowering operator. The sum rules thus obtained are always compatible with the Gell-Mann-Okubo mass splittings of hadrons. They exhibit a simple modification of exact-SU(3) sum rules, but the effect is, in general, quite significant. As a specific application, the Y0*(1405) transition is discussed in some detail in order to compare with the recent result of Gell-Mann, Oakes, and Renner (GOR), based on a different approximation for broken SU(3) symmetry. It is shown that, in the absence of singlet-octet mixing, both approaches give the same result in this particular case, and that the GOR approximation can, in fact, be derived from our asymptotic SU(3) symmetry. Our asymptotic SU(3) symmetry, however, appears to be a more general and far-reaching prescription, useful in broken SU(3) symmetry when combined with the use of equal-time commutation relations involving the charge VK. A comment is also made about the hard-kaon and η-meson extrapolation resulting from the use of kaon and η-meson PCAC. The result is applied to the derivation of the values of the ΛNK and ΣNK couplings from the experimental information on the axial-vector semileptonic couplings of hyperons. The result is consistent with experiment.

Asymptotic su(3) symmetry and baryon couplings.

Broken SU(3) sum rules involving the baryon resonances with J = 12, J = 32 and J = 52

General intermultiplet mass formulas between the decuplet and octet baryons are derived, based on the chiral $\mathrm{SU}(3)\ensuremath{\bigotimes}\mathrm{SU}(3)$ charge algebra and the asymptotic $\mathrm{SU}(3)$ symmetry imposed only on the charge operator ${V}_{K}$ which is the $\mathrm{SU}(3)$ raising or lowering operator in the symmetry limit. In the absence of particle mixing, the formulas take on a simple form and are useful as a first guide in deducing baryon mass spectra. Inclusion of particle mixing is possible. The formulas imply that the equal squared-mass spacings of decuplet states ${({\ensuremath{\delta}}_{a})}^{2}$ ($a$ specifies the quantum numbers of the $a$ decuplet) are universal [i.e., ${({\ensuremath{\delta}}_{a})}^{2}=\mathrm{const}$] and, furthermore, equal to the universal spacings of the $\ensuremath{\Xi}$ and $\ensuremath{\Sigma}$ members of any octet baryons. In the case of the usual ${\mathrm{\textonehalf{}}}^{+}$ and ${\frac{3}{2}}^{+}$ baryons, the formulas coincide with the well-known $\mathrm{SU}(6)$ mass formula. Broken-$\mathrm{SU}(3)$ sum rules (in the absence of mixing) for general transitions of the form $(\mathrm{octet}\mathrm{baryon})\ensuremath{\rightarrow}(\mathrm{octet}\mathrm{baryon})+\ensuremath{\pi} (or l+\overline{\ensuremath{\nu}})$ are also obtained. In particular, the strong decays, ${\frac{5}{2}}^{\ensuremath{-}}\ensuremath{\rightarrow}{\frac{1}{2}}^{+}+\ensuremath{\pi}$ and ${\frac{5}{2}}^{+}\ensuremath{\rightarrow}{\frac{1}{2}}^{+}+\ensuremath{\pi}$, are discussed in detail. The sum rules, in general, give rise to significant $\mathrm{SU}(3)$-breaking effects. However, for the familiar axial-vector ${\frac{1}{2}}^{+}\ensuremath{\rightarrow}{\frac{1}{2}}^{+}+l+\overline{\ensuremath{\nu}}$ transitions, our broken-$\mathrm{SU}(3)$ sum rules assume the same forms as satisfied by the hypothetical exact $\mathrm{SU}(3)$ couplings. This justifies the use of the original Cabibbo analysis [in broken-$\mathrm{SU}(3)$ symmetry] in determining the value of the axial-vector Cabibbo angle.

S. Matsuda

Papers

Current algebra, infinite momentum and sum rules