The Physics of $K^0-{\bar{K}}^0$ Mixing: ${\hat{B}}_K$ and $\Delta M_{LS}$ in the Chiral Quark Model

Abstract

We compute the ${\hat{B}}_K$ parameter and the mass difference $\Delta M_{LS}$ of the $K^0-{\bar{K}}^0$ system by means of the chiral quark model. The chiral coefficients of the relevant $\Delta S=2$ and $\Delta S=1$ chiral lagrangians are computed via quark-loop integration. We include the relevant effects of one-loop corrections in chiral perturbation theory. The final result is very sensitive to non-factorizable corrections of $O({\alpha }_{S}N)$ coming from gluon condensation. The size of the gluon condensate is determined by fitting the experimental value of the amplitude $K^{+}\to {\pi }^{+}{\pi }^0$. By varying all the relevant parameters we obtain ${\hat{B}}_K=0.87\pm 0.33.$ We evaluate within the model the long-distance contributions to $\Delta M_{LS}$ induced by the double insertion of the $\Delta S=1$ chiral lagrangian and study the interplay between short- and long-distance amplitudes. By varying all parameters we obtain $\Delta M_{LS}^{th}/\Delta M_{LS}^{exp}=0.76_{-0.34}^{+0.64}.$ Finally, we investigate the phenomenological constraints on the Kobayashi-Maskawa parameter Im ${\lambda }_t$ entering the determination of ${ϵ}^{\prime }/ϵ$.

Comments: 31 pages, Latex file including 7 eps figures. Revised version to appear in Nucl. Phys. B