Current-algebra theorems forKπscattering

Abstract

On-mass-shell current-algebra sum rules for $K\ensuremath{\pi}$ scattering are investigated. At threshold these sum rules relate the combinations ${a}^{\frac{1}{2}}\ensuremath{-}{a}^{\frac{3}{2}}$ and ${a}^{\frac{1}{2}}+{2a}^{\frac{3}{2}}$ of $I=\frac{1}{2} \mathrm{and} \frac{3}{2}$ $S$-wave scattering lengths to their soft-meson predictions plus "correction" terms. A dispersion-theoretic approach is used to calculate the corrections to both the soft-koan and soft-pion values for each of these scattering length combinations. The dispersion relations are assumed to be dominated by the ${K}^{*}(890)$ and a ${J}^{P}={0}^{+}\ensuremath{\kappa}$ meson. The correction to the soft-pion value ${a}^{\frac{1}{2}}\ensuremath{-}{a}^{\frac{3}{2}}=0.210{{m}_{\ensuremath{\pi}}}^{\ensuremath{-}1}$ is found to be very small, thus suggesting that the on-mass-shell current-algebra prediction for ${a}^{\frac{1}{2}}\ensuremath{-}{a}^{\frac{3}{2}}$ is nearly equal to the soft-pion value. Self-consistency requirements of the analysis favor $\frac{{F}_{K}}{{F}_{\ensuremath{\pi}}}\ensuremath{\simeq}1.22$; this implies that a fairly large (\ensuremath{\sim}30%) correction to the corresponding soft-koan prediction of ${a}^{\frac{1}{2}}\ensuremath{-}{a}^{\frac{3}{2}}\ensuremath{\approx}0.14{{m}_{\ensuremath{\pi}}}^{\ensuremath{-}1}$ is needed. The ${K}^{*}$ and $\ensuremath{\kappa}$ contributions to the latter correction increase this value to $0.177{{m}_{\ensuremath{\pi}}}^{\ensuremath{-}1}$. Furthermore, since the ${K}^{*}$ and $\ensuremath{\kappa}$ corrections to and the soft-koan value of ${a}^{\frac{1}{2}}\ensuremath{-}{a}^{\frac{3}{2}}$ decrease with increasing $\frac{{F}_{K}}{{F}_{\ensuremath{\pi}}}$, a value of $\frac{{F}_{K}}{{F}_{\ensuremath{\pi}}}\ensuremath{\gtrsim}1.3$ seems, at least in the context of the present study, to be rather unlikely.