Low-EnergyK−NScattering andSU(3)Invariance

Abstract

Phase shifts from $K_{+}-p$ and $K_{+}-d$ scattering up to $800 \frac{\mathrm{MeV}}{c}$ are fitted by a sum of single-particle exchange terms and a "background" term representing unknown short-range forces. The background is described by a low-order polynomial in $s$ and $t$ in the invariant amplitudes. The single-particle states in the $u$-channel are $\Lambda$, $\Sigma$, $Y_1^{}{}_{}{}^{*}$(1385 MeV), $Y_0^{}{}_{}{}^{*}$(1405 MeV), $Y_1^{}{}_{}{}^{*}$(1520 MeV), and $Y_1^{}{}_{}{}^{*}$(1660 MeV). A continuum of $u$-channel $\overline{K}-N$ scattering below $400 \frac{\mathrm{MeV}}{c}$ is also included. In the $t$ channel the states are $\rho$ and a fictitious particle $\chi$ which is supposed to represent the average effect of $\omega$ and $\varphi$. The $Y^{*}N\overline{K}$ coupling constants are known from experiment or a combination of theory and experiment. The ratio of vector to tensor $\rho \mathrm{NN}$ coupling constants is taken from nucleon electromagnetic-structure data. The other coupling constants are regarded as free parameters to be determined by fitting $K-N$ data. For good fits it is necessary to retain the first three orders in the expansion of the background term. This entails eight free parameters. Two sets of fits to the data are found, one with the Fermi-type $I=0\mathrm{}$ phase shifts (${\delta }_{\frac{p3\mathrm{}}{2}}$ large and positive) and one with Yang-type $I=0\mathrm{}$ phases (${\delta }_{\frac{p1\mathrm{}}{2}}$ large and positive). In the Fermi case the $\Sigma$ and $\rho$ coupling constants each differ by more than five standard deviations from predictions based on $\mathrm{SU}\left(3\right)\mathrm{}$ symmetry. The $\Sigma$ constant also disagrees badly with results from forward-angle dispersion relations. In the case of the Yang-type phase shifts, the coupling constants are essentially in agreement with $\mathrm{SU}\left(3\right)\mathrm{}$ predictions. In the Yang case the $\rho$ term is necessary for an acceptable fit, in spite of the large number of parameters. The parameters for $\chi$ exchange are badly determined by the data, so there is no possibility of working with $\omega$ and $\varphi$ coupling constants as separate free parameters. Exchange of scalar particles is considered as a model for the effects of "ABC" or "$\sigma$" $\pi -\pi$ interactions at 310 or 400 MeV, and the $K_1-K_1$ interaction at 1000 MeV. Inclusion of such exchange terms affects the parameter values of the other terms only slightly, and therefore does not change the qualitative character of the fits to the data. The $Y^{*}$ terms can also be omitted without changing the general character of the fits. They are not small, but they are well represented by the background terms. Unitarity corrections are estimated to be rather minor; they do not change the main conclusions.