New prediction for the directCP-violating parameterɛ′/ɛand theΔI=1/2rule

The low-energy dynamics of QCD is investigated with special attention paid to the matching between QCD and chiral perturbation theory (ChPT), and also to some useful algebraic chiral operator relations which survive even when we include chiral loop corrections. It then allows us to evaluate the hadronic matrix elements below the energy scale ${\Lambda }_{\chi }\simeq 1\mathrm{GeV}.$ Based on the new analyses, we present a consistent prediction for both the direct CP-violating parameter ${\varepsilon }^{\prime }/\varepsilon$ and the $\Delta I=1/2\mathrm{}$ rule in kaon decays. In the leading ${1/N}_c$ approximation, the isospin amplitudes $A_0$ and $A_2$ are found to agree well with the data, and the direct CP-violating parameter ${\varepsilon }^{\prime }/\varepsilon$ is predicted to be large, which also confirms our earlier conclusion. Its numerical value is ${\varepsilon }^{\prime }/\varepsilon {=23.6\mathrm{}}_{-7.8}^{+12.4\mathrm{}}\times {10}^{-4}(\mathrm{Im}{\lambda }_t/1.2\mathrm{}\times {10}^{-4})$ which is no longer sensitive to the strange quark mass due to the matching conditions. Taking into account a simultaneous consistent analysis on the isospin amplitudes $A_0$ and $A_2,$ the ratio ${\varepsilon }^{\prime }/\varepsilon$ is in favor of the values ${\varepsilon }^{\prime }/\varepsilon =(20\mathrm{}\pm 9)\times {10}^{-4}.$