AsymptoticSU(3)Symmetry and Baryon Couplings

Abstract

General broken-$\mathrm{SU}(3)$ sum rules for baryon transitions ${B}^{\ensuremath{'}}\ensuremath{\rightarrow}{(\frac{1}{2})}^{\ifmmode\pm\else\textpm\fi{}}+P$ (${B}^{\ensuremath{'}}$ denotes a nonet or octet baryon with arbitrary spin and parity) are derived. The approach is based on the use of a chiral $\mathrm{SU}(3)\ensuremath{\bigotimes}\mathrm{SU}(3)$ charge algebra, the hypothesis of partially conserved axial-vector current (PCAC) and, in particular, the assumption of asymptotic $\mathrm{SU}(3)$ symmetry formulated for the matrix elements of the vector charge ${V}_{K}$. The ${V}_{K}$ is the $\mathrm{SU}(3)$ 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-$\mathrm{SU}(3)$ sum rules, but the effect is, in general, quite significant. As a specific application, the ${{Y}_{0}}^{*}(1405)$ 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 $\mathrm{SU}(3)$ 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 $\mathrm{SU}(3)$ symmetry. Our asymptotic $\mathrm{SU}(3)$ symmetry, however, appears to be a more general and far-reaching prescription, useful in broken $\mathrm{SU}(3)$ symmetry when combined with the use of equal-time commutation relations involving the charge ${V}_{K}$. A comment is also made about the hard-kaon and $\ensuremath{\eta}$-meson extrapolation resulting from the use of kaon and $\ensuremath{\eta}$-meson PCAC. The result is applied to the derivation of the values of the $\ensuremath{\Lambda}\mathrm{NK}$ and $\ensuremath{\Sigma}\mathrm{NK}$ couplings from the experimental information on the axial-vector semileptonic couplings of hyperons. The result is consistent with experiment.