Dispersive Sum-Rule Approach toπ−KScattering

Abstract

The dispersive sum-rule method, originally developed by Fubini and Furlan, is applied to $\ensuremath{\pi}\ensuremath{-}K$ elastic scattering. Sum rules are derived for the $I=\frac{1}{2}$ and $I=\frac{3}{2}$ scattering amplitudes, and the isospin-antisymmetric combination of the $s$-wave scattering lengths is calculated. These expressions contain terms involving the off-mass-shell kappa-kaon-pion coupling constant. By following a procedure introduced by Dashen and Weinstein, and assuming that the $\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(3)$ symmetry-breaking part of the strong-interaction Hamiltonian transforms according to the $(3,{3}^{*})+({3}^{*},3)$ representation of $\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(3)$, we evaluate the off-shell corrections. In the evaluation of the off-shell corrections, we obtain expressions for the postulated $\ensuremath{\kappa}$-meson mass and decay width, consistent with a recent experimental indication. The $s$-wave scattering lengths are consistent with other current-algebra and phenomenological-Lagrangian calculations, but smaller than those recently reported from calculations based on the leading term Veneziano model.