Decuplet Sum Rules for Pseudoscalar-Meson-Hadron Scattering in Exact and BrokenSU(3)Symmetry

Abstract

It is assumed that an arbitrary $s$-channel amplitude $A+B\ensuremath{\rightarrow}C+D$ superconverges when the $t$-channel system $A\overline{C}$ belongs to the 10 (or $\overline{1}\overline{0}$) representation of $\mathrm{SU}(3)$ or has the quantum numbers $I=\frac{3}{2}$, $Y=\ifmmode\pm\else\textpm\fi{}1$ or $I=0$, $Y=\ifmmode\pm\else\textpm\fi{}2$. The particular cases of $\mathrm{PP}$, $\mathrm{PV}$, and $\mathrm{PB}$ elastic scattering are studied, both in the $\mathrm{SU}(3)$- symmetric and in the broken-symmetry limit. The $\mathrm{PP}$ sum rule is saturated in the $\mathrm{SU}(3)$ limit by the nonet spectrum ${J}^{\mathrm{PC}}={1}^{\ensuremath{-}\ensuremath{-}},{2}^{++}$. The Okubo form for the nonet couplings is obtained in the limit of nonet mass degeneracy. When $\mathrm{SU}(3)$ symmetry is relaxed, the resultant sum rules provide a constraint on the form (but not the scale) of symmetry-breaking effects. An unphysical result (implying a complex mixing angle) derived by Sakurai from the Weinberg second sum rule and a vector-dominance assumption is reproduced by the vector saturation of a $\overline{K}K$ backward scattering amplitude. It is pointed out that the effects of higher-spin mesons can, in this approach, restore the reality of the mixing angle. Analogous sum rules for $\mathrm{PV}$ elastic scattering imply the $\mathrm{SU}(6)$ result for the singlet/octet $\mathrm{PVV}$ coupling ratio. In the limit of $\mathrm{SU}(3)$ symmetry, baryon octet-decouplet mass degeneracy, and zero pseudoscalar-meson mass, the $\mathrm{PB}$ sum rule, when combined with the sum rule of Sakita and Wali, yields the $\mathrm{SU}(6)$ result for the $\frac{f}{d}$ ratio and the decuplet/octet coupling ratio, independent of experimental input. When $\mathrm{SU}(3)$ symmetry is relaxed, the resultant $\mathrm{PB}$ sum rules, one of which has been evaluated by Chan and Yen, can provide information on the $\mathrm{BBP}$ coupling constants. Typically, the sum rules studied, based initially only on the assumptions of $\mathrm{SU}(3)$ symmetry and superconvergence, lead to the results of a higher symmetry when the degenerate mass spectrum of the higher symmetry is used in the saturation scheme. When the mass degeneracy is broken, the sum rules yield information on how this higher symmetry is broken. When $\mathrm{SU}(3)$ symmetry is relaxed, the sum rule splits into a family of sum rules which are classified according to $t$-channel hypercharge and isospin content. Some of these sum rules are still valid provided certain weaker assumptions (such as the conservation, or the absence, of appropriate currents) continue to hold, and these sum rules, in turn, provide information on $\mathrm{SU}(3)$ violations.