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

Abstract

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.

Comments: Latex2e, JHEP.cls, 32 pages, 9 figures. Short-distance analysis updated. Parts of Sects. 3 and 6 revised