Possible One-Channel Castille jo-Dalitz-Dyson Poles inπNandππScattering

We explore interchannel coupling as a qualitative guide in dynamical calculations, and as an ingredient in models of $\pi N$ and $\pi \pi$ scattering. Our first model is the $\mathrm{SU}\left(6\right)\mathrm{}$-symmetric static baryon bootstrap of Capps, Belinfante, and Cutkosky. We find, on the assumption of a successful bootstrap, that the $P_{11}$ and $P_{33}\pi N$ waves each contain a one-channel Castillejo-Dalitz-Dyson (CDD) pole. If these CDD poles survive $\mathrm{SU}\left(6\right)\mathrm{}$ symmetry breaking, we can understand and correlate a large number of heretofore puzzling features of these waves, for example, the well-established zero in the $P_{11}$ amplitude, and the observed $P_{33}$ high-energy phase shift. Secondly, we assume that the observed nonet of $2^{+}$ particles is the Regge recurrence of a nonet of $0^{+}$ extinct bound states, and that these latter can be calculated in a dynamical $0^{-}$ $0^{-}$ (3-channel) calculation. We find, in exact $\mathrm{SU}\left(3\right)\mathrm{}$ symmetry, that certain waves contain one-channel CDD poles; for example, the $T=J=0\mathrm{}\pi \pi$ wave contains one. On the assumption of $\mathrm{SU}\left(6\right)\mathrm{}$, however, every extinct bound state appears to bring down a CDD pole, in which case there would be no observable consequence of the $0^{+}$ nonet. In these models, we try to avoid quantitative group theory, emphasizing general principles that we hope may be valid in a larger context.