Study of a Self-Consistent Calculation of theΩ−as aK¯ΞBound State

Abstract

The $\overline{K}\Xi {\Omega }^{-}$ system has been studied in a self-consistent manner, both to see if the ${\Omega }^{-}$ can be interpreted as a $T=0\mathrm{}\overline{K}\Xi$ bound state, and to study such calculations in a situation where only one two-particle channel need be considered. Wefollow closely the method of Singh and Udgaonkar, and Pati, using for dynamical singularities both a short cut due to single particle exchange and a self-consistently determined distant left cut. Only the $J={\frac{3}{2}}^{+}$ partial wave is considered. The main result is a curve relating the strength (called $g^2$) of the single-particle exchange-forces to the position of a bound or resonant state. The binding energy is found to increase with increasing $g^2$ until the mass of the bound state is about 1680 MeV. Further increases in $g^2$ have essentially no effect. On the other hand, it is found that resonant states exist for $g^2\le 0$, so that the calculation as it stands allows resonances with repulsive single-particle exchange forces. The calculation in its present form is consistent with bound or resonant states for both $T=0\mathrm{}$ and $T=1\mathrm{}$, the former at about 1680 MeV for reasonable values of $g^2$. It is shown that the results are essentially independent of subtraction point, matching points, and similar parameters. The dependence of the results on the short cut and on the distant cut is discussed. The manner in which calculations using only the Born approximation for the dynamical singularities can give the same results as ours is indicated. The possible existence of two zeros in Re $D$, noticed by Abers and Zachariasen, is briefly discussed. A second zero does appear in the present calculation at about 4 GeV., but results from the phase shift decreasing through $\frac{1}{2}\pi$ rather than from the Abers-Zachariasen mechanism. Thus, no alternative solution is found in the present calculation.