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 value of the quark and gluon condensates and the constituent quark mass. We take these as input parameters to be fixed by fitting the $\Delta I =1/2$ selection rule of the isospin I=0 and I=2 nonleptonic decay amplitudes. Such a fit gives values for the condensates very 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 $\hat B_K = 1.3 ^{+ 0.2}_{-0.1}$. This rather large value corresponds to a $O(p^4)$ correction of about 45%; it is in agreement with the determination from the experimental value of $\Delta M_{LS}$ from which we find $\hat B_K = 1.2 ^{+ 0.2}_{-0.1}$ after having included long- and short-distance contributions.