Scalar Form Factors just beyond the Tree Approximation

Using the algebra of currents and current divergences derived from the chiral $\mathrm{SU}\left(3\right)\mathrm{}$ symmetry-breaking model of Gell-Mann, Oakes, and Renner and a method of pole and cut dominance, we derive two-parameter effective-range formulas for scalar $(I=J=\frac{1}{2})K\pi$ and $\mathrm{}(I=J=)\mathrm{}\pi \pi$ form factors. The fit to the $I=\frac{1}{2}$ $s$-wave $K\pi$ phase-shift data of Antich et al. favors their "lower" solution with $m_K=1150\mathrm{}$ MeV and supports the existence of a broad $\kappa$ resonance. It is shown that the introduction of the similarly broad $\epsilon$ resonance which emerges from these considerations into the weak-interaction scheme ${K_S}^0\to \epsilon \to \pi \pi$ makes agreement with the experimental value of the ${K_L}^0-{K_S}^0$ mass difference difficult, if the ${K_L}^0$ mass shift is not small.