The K → ππ decays in the large-N limit quark evolution

We use the large-N limit to obtain a consistent picture of the K → ππ amplitudes, A(K → ππ). We emphasize that the full evaluation of A(K → ππ) should include two contributions: (i) the evolution from M_W down to μ ≅ O(1 GeV) done in the quark picture and (ii) the evolution from μ ≅ O(1 GeV) down to low energy O(λ_QCD, m_π, m_K) which must be done in the meson picture. Using chiral perturbation theory at the tree level, we confirm that the matrix elements of the dominant Q₆ penguin operator are enhanced by an order of magnitude over those of the current-current operators. We identify an additional (with respect to standard estimates) enhancement of the penguin contributions to the real parts of A(K → ππ) relevant for the ΔI = 1 2 rule, which comes from an imcomplete GIM cancellation above the charm quark mass. The corresponding contributions to the imaginary parts and to the ratio ε' ε are found to be slightly suppressed. Performing the short distance analysis to first nontrivial order in 1/N, we demonstrate its validity and simplicity. We obtain encouraging results for A(K → ππ) amplitudes and point out that further improvements can be obtained through the inclusion of the low energy meson evolution which is presented in a separate publication. We find ε′ ε ≅ 5 · 10^-3 for m_t = 40 GeV.