Factorization Approach for the $\Delta I=1/2$ Rule and $\epsilon'/\epsilon$ in Kaon Decays

TL;DR

This paper investigates the $ riangle I=1/2$ rule and direct CP violation $ ext{Re}(rac{ ext{epsilon}'}{ ext{epsilon}})$ in kaon decays using an effective Hamiltonian and generalized factorization, providing insights into hadronic matrix elements and scheme dependence.

Contribution

It introduces a scheme- and scale-independent approach to hadronic matrix elements in kaon decays, improving upon chiral methods and calculating bag parameters with scheme and mass dependence.

Findings

A0/A2 ratio of 13-15 for specific strange quark masses.

Calculated $ ext{Re}(rac{ ext{epsilon}'}{ ext{epsilon}})$ in the range $(0.7-1.6) imes 10^{-3}$.

Bag parameters $B_8^{(2)}$ and $B_6^{(0)}$ are nearly scheme independent.

Abstract

The $\Delta I=1/2$ rule and direct CP violation $\epsilon'/\epsilon$ in kaon decays are studied within the framework of the effective Hamiltonian approach in conjunction with generalized factorization for hadronic matrix elements. We identify two principal sources responsible for the enhancement of A0/A_2: the vertex-type as well as penguin-type corrections to the matrix elements of four-quark operators, which render the physical amplitude renormalization-scale and -scheme independent, and the nonfactorized effect due to soft-gluon exchange, which is needed to suppress the $\Delta I=3/2$ $K\to\pi\pi$ amplitude. Contrary to the chiral approach which is limited to light meson decays and fails to reproduce the A2 amplitude, the aforementioned approach for dealing with scheme and scale issues is applicable to heavy meson decays. We obtain A0/A2=13-15 if $m_s$(1 GeV) lies in the range (125-175)MeV. The bag parameters $B_i$, which are often employed to parametrize the scale and scheme dependence of hadronic matrix elements, are calculated in two different renormalization scehemes. It is found that $B_8^{(2)}$ and $ B_6^{(0)}$, both of order 1.5 at $\mu=1$ GeV, are nearly $\gamma_5$ scheme independent, whereas $B^{(0)}_{3,5,7}$ as well as $B_7^{(2)}$ show a sizable scheme dependence. Moreover, only $B_{1,3,4}^{(0)}$ exhibit a significant $m_s$ dependence, while the other $B$-parameters are almost $m_s$ independent. For direct CP violation, we obtain $\epsilon'/\epsilon=(0.7-1.1)\times 10^{-3}$ if $m_s(1 {\rm GeV})=150$ MeV and $\epsilon'/\epsilon=(1.0-1.6)\times 10^{-3}$ if $m_s$ is as small as indicated by some recent lattice calculations.