Cutkosky-Leon Bootstrap Model and the Strong-Coupling Model

Abstract

The static $\ensuremath{\pi}\ensuremath{-}N$ bootstrap model based on the Bethe-Salpeter equation is applied to the study of the mass spectrum and the meson coupling constants when isobars heavier than $N_{3,3}^{}{}_{}{}^{*}$ are included. By assuming $I=J$, we have obtained the solutions for two models which are constructed by adding, up to $I=J=\frac{7}{2}$, the isobars $N_{2I,2J}^{}{}_{}{}^{*}$ to those of the familiar model of the $N\ensuremath{-}N_{3,3}^{}{}_{}{}^{*}$. We discover also that this Bethe-Salpeter bootstrap model (or Cutkosky-Leon model) contains a solution which is identical to that from the strong-coupling model for the case of charge-symmetric pseudoscalar-meson theory; in this solution there are an infinite number of degenerate isobars with their isospin equal to their spin. Also included in the table of results is the $\mathrm{SU}(4)$ prediction for the various coupling constants when heavier isobars are included. The $\mathrm{SU}(4)$ model contains in addition to the pions the $\ensuremath{\rho}$ and $\ensuremath{\eta}$ mesons. We notice that the results of bootstrapping and $\mathrm{SU}(4)$ are similar and that the deviations between them are less than 50%. It is also noticed that the heavier isobars act in the bootstrap model like a small perturbation on the low-lying isobar states.