The Delta I = 1/2 Rule and B_K at O(p^4) in the Chiral Expansion

We calculate the hadronic matrix elements to $O(p^4)$ in the chiral expansion for the ($\Delta S =1$) $K^0 \to 2 \pi$ decays and the ($\Delta S=2$) $\bar K^0$-$K^0$ oscillation. This is done within the framework of the chiral quark model. The chiral coefficients thus determined depend on the values of the quark and gluon condensates and the constituent quark mass. We show that it is possible to fit the $\Delta I =1/2$ rule of kaon decays with values of the condensates close to those obtained by QCD sum rules. The renormalization invariant amplitudes are obtained by matching the hadronic matrix elements and their chiral corrections to the short-distance NLO Wilson coefficients. For the same input values, we study the parameter $\hat B_K$ of kaon oscillation and find $\hat B_K = 1.1 \pm 0.2$. As an independent check, we determine $\hat B_K$ from the experimental value of the $K_L$-$K_S$ mass difference by using our calculation of the long-distance contributions. The destructive interplay between the short- and long-distance amplitudes yields $\hat B_K = 1.2 \pm 0.1$, in agreement with the direct calculation.