Penguin diagrams in theΔI=1/2rule andε′/εwith σ models

We study the correlation between ${\epsilon }^{\prime }/\epsilon$ and the $\Delta I=1/2\mathrm{}$ rule in the frame-work of the nonlinear σ model including scalar mesons. Using this model we estimate the chiral corrections by resumming to all orders in chiral perturbation theory, the contribution of a class of diagrams within the factorization approximation. With these matrix elements and changing the scalar meson mass, we find that there is a correlation between ${\epsilon }^{\prime }/\epsilon$ and the $\Delta I=1/2\mathrm{}$ amplitude. However, it is difficult to explain both ${\epsilon }^{\prime }/\epsilon$ and the $\Delta I=1/2\mathrm{}$ amplitude simultaneously. In order to be compatible with ${\epsilon }^{\prime }/\epsilon ,$ typically, about half of the $\Delta I=1/2\mathrm{}$ amplitude can be explained at most. Our result suggests there may be a substantial nonfactorizable contribution to CP conserving $\overrightarrow{K}\pi \pi$ amplitudes.