The K ---> pi pi 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 MW down to μ congruent with O(1 GeV) done in the quark picture and (ii) the evolution from μ congruent with O(1 GeV) down to low energy O(λQCD, mπ, mK) 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 Image rule, which comes from an imcomplete GIM cancellation above the charm quark mass. The corresponding contributions to the imaginary parts and to the ratio var epsilon'/var epsilon 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 var epsilon′/var epsilon congruent with 5 · 10−3 for mt = 40 GeV.