Dynamical Calculation of the32+Resonance Spectrum andSU(3)Symmetry Breaking

Abstract

The multichannel relativistic Schr\"odinger equation is solved for the ${\frac{3}{2}}^{+}$ partial-wave amplitude with an energy-dependent potential obtained by computing the baryon exchange contribution to pseudoscalarmeson-baryon scattering. Since a cutoff parameter is not needed in our calculations, we use the coupling constant as an adjustable parameter. Thus we take $\frac{{g}^{2}}{4\ensuremath{\pi}}=38$ in order to obtain the ${N}^{*}(1236)$ at the correct experimental energy. The model yields the usual ${\frac{3}{2}}^{+}$ decuplet and predicts in addition the existence of a 27-dimensional $\mathrm{SU}(3)$ representation [to which ${P}_{33}(1690)$, ${P}_{13}(1860)$, ${Z}_{1}(1900)$, and some higher ${\ensuremath{\Sigma}}^{*}$, ${\ensuremath{\Xi}}^{*}$, and ${\ensuremath{\Lambda}}^{*}$ resonances could belong] as well as a second decuplet at much higher energies. Certain resonances ("exotic" ones with $I=2$, $Y=0$; $I=\frac{3}{2}$, $Y=\ensuremath{-}1$; and $I=1$, $Y=\ensuremath{-}2$) are very broad (\ensuremath{\approx} 1000 MeV); others (the usual ones including the exotic $I=1$, $Y=2$ $\mathrm{KN}$ resonance) have a width \ensuremath{\le}400 MeV. The dependence of the resonance spectrum on the breaking of $\mathrm{SU}(3)$ symmetry as well as on the coupling constant and on the $\frac{F}{D}$ ratio is discussed. A value of the $\frac{F}{D}$ ratio of about 0.4 seems the best fit to experiments. The results are compared with the ${\frac{3}{2}}^{+}$ predictions of other models such as $\frac{N}{D}$ calculations, quark model, strong-coupling theory, and ${\mathrm{SU}(3)}^{\ensuremath{'}\ensuremath{'}}$ symmetry. In the two last-named models, a 27 appears also as a higher super-multiplet, which is forbidden in the quark model. Additional octets and $\overline{1}\overline{0}$ representations [predicted also by ${\mathrm{SU}(3)}^{\ensuremath{'}\ensuremath{'}}$ and the quark model] are obtained as unphysical objects in our calculation.