Test of Global Symmetry inK−−pReactions

A method of testing the hypothesis that there is global symmetry of the pion-baryon interaction is proposed: Upon analyzing low-energy $K^{-}-p$ scattering data, one finds a variety of scattering length solutions which are compatible with the elastic scattering, charge exchange scattering, and total charged hyperon production. Our suggestion involves the use of the experimental $\frac{{\Sigma }^{+}}{{\Sigma }^{-}}$ ratios (a) to test the global symmetry hypothesis (or any other quantitative description of the pion-hyperon interaction), (b) to reduce the ambiguity in the $\overline{K}-N$ scattering length solutions, (c) hence, to predict the ${\Sigma }^0$ cross section. We need to know the $\pi -Y$ phase shifts, in addition to the $\overline{K}-N$ scattering lengths in order to predict the ${\Sigma }^{+}$, ${\Sigma }^{-}$ cross sections. If there is global symmetry of pion-baryon interactions, then we know the $\pi -Y$ phases in the absence of a $\overline{K}-N$ reaction channel. We demonstrate how the actual $\pi -Y$ phase shifts can be obtained, in a nonperturbative manner, from idealized $\pi -Y$ phases (i.e., in the absence of $\overline{K}-N$ reactions) and the $\overline{K}-N$ scattering lengths. Earlier proposals making use of the dependence of the hyperon production ratios on the $\pi -Y$ phase shifts are also examined in terms of this result. Certain fits to the present rough data with scattering lengths of negative real part are shown to be incompatible with global symmetry. The proposed analysis involves the assumption that the $\overline{K}$ is an isotopic doublet.