Effect of Closed Inelastic Channels on the Width of Resonances

The effects of including nearby inelastic channels are examined on three separate examples of $p$-wave resonances: (a) a three-channel ($\pi \pi$, $\pi \omega$, $K\overline{K}$) system is treated in a calculation of the $\rho$ (760 MeV) resonance, (b) a two-channel ($\pi K$, $\eta K$) system is considered for the $K^{*}\mathrm{}\left(885\right)\mathrm{}$ and, (c) a two-channel ($\pi N$, $K\Sigma$) system is treated for the $N^{*}\mathrm{}\left(1238\right)\mathrm{}$. The input forces in (a) and (b) are taken to be due to the exchange of the known vector mesons and in (c) the baryons alone. The masses and coupling constants of the exchanged particles are taken from experiment; the coupling constants which cannot be related in a direct manner to experimental quantities are related to the others by the assumption of $S{U}_3$ symmetry. Straight cutoffs are used to insure that the multichannel $N{D}^{-1\mathrm{}}$ equations have a unique solution. Full numerical solutions of these equations are obtained by the matrix inversion technique. The values of the cutoffs were adjusted in the three examples to yield a peak in the cross section at the observed energy for the one-channel case and readjusted (to a lower value) to reproduce the resonant energy in the presence of the inelastic channels. This allows one to study the effect of the additional channels on the width and shape of the resonances. With the latter values of the cutoffs, the inelastic channels are turned off and the changes in the positions of the resonances are noted in order to estimate the relative contributions of the forces producing the resonance. It is noted, e.g., that the $\pi \omega$ channel has more effect on the position of the $\rho$ than does the $K\overline{K}$ channel; also, the $K\Sigma$ channel has relatively little effect on the $N^{*}$ (even though for pure $S{U}_3$ symmetry the $N^{*}$ is equally composed of $\pi N$ and $K\Sigma$). All the calculated widths are two to five times larger than the observed values, indicating that the lowest mass channels coupled via the longest range forces do not constitute a realistic model.