Penguins diagrams in Delta I=1/2 rule and \epsilon'/\epsilon with \sigma models

TL;DR

This paper investigates the relationship between the direct CP violation parameter '/ and the I=1/2 rule in kaon decays using a non-linear sigma model with scalar mesons, highlighting the challenges in simultaneously explaining both phenomena.

Contribution

It introduces a correlation analysis between '/ and I=1/2 rule within a non-linear sigma model including scalar mesons, emphasizing the role of scalar meson mass variation.

Findings

Correlation exists between '/ and I=1/2 amplitude.

Difficulty in explaining both '/ and I=1/2 simultaneously.

Substantial non-factorizable contributions may be needed.

Abstract

We study the correlation between $\epe$ and $\Delta I=1/2$ rule in the framework of non-linear $\sigma$ model including scalar mesons. The matrix elements of QCD and Electroweak (EW) penguin operators are computed within factorization approximation. Using the matrix elements and changing the scalar meson mass, we find that there is correlation between $\epe$ and $\Delta I=1/2$ amplitude. However, it is difficult to explain both $\epe$ and $\Delta I=1/2$ amplitude simultaneously. In order to be compatible with $\epe$, typically, about half of $\Delta I=1/2$ amplitude can be explained at most. Our result suggests there may be substantial non-factorizable contribution to CP conserving $K \to \pi \pi$ amplitudes.