Abstract
We demonstrate that in the large N approach developed by the authors in collaboration with Bardeen, the parameters B_6^{(1/2)} and B_8^{(3/2)} parametrizing the K\to\pi\pi matrix elements <Q_6>_0 and <Q_8>_2 of the dominant QCD and electroweak operators receive both negative O(1/N) corrections such that B_6^{(1/2)} < B_8^{(3/2)}<1 in agreement with the recent lattice results of the RBC-UKQCD collaboration. We also point out that the pattern of the size of the hadronic matrix elements of all QCD and electroweak penguin operators Q_i contributing to the K\to \pi \pi amplitudes A_0 and A_2, obtained by this lattice collaboration, provides further support to our large N approach. In particular, a very precise lattice result for the matrix element <Q_8>_0 implies for the corresponding parameter B_8^{(1/2)}=1.0\pm 0.2 to be compared with large N value B_8^{(1/2)}=1.1\pm 0.1. We discuss briefly the implications of these findings for the ratio epsilon'/epsilon. In fact, with the precise value for B_8^{(3/2)} from RBC-UKQCD collaboration, our upper bound on B_6^{(1/2)} implies epsilon'/epsilon in the SM roughly by a factor of two below its experimental value (16.6\pm 2.3)\times 10^{-4}. We also briefly comment on the parameter \hat B_K and the \Delta I=1/2$ rule.
Highlights
ReA0 and ReA2 in the ballpark of experimental values [10]
We demonstrate that in the large N approach developed by the authors in collaboration with Bardeen, the parameters B6(1/2) and B8(3/2) parametrizing the K → ππ matrix elements Q6 0 and Q8 2 of the dominant QCD and electroweak operators receive both negative O(1/N ) corrections such that B6(1/2) ≤ B8(3/2) < 1 in agreement with the recent lattice results of the RBC-UKQCD collaboration
We have compared the structure of the hadronic matrix elements in K → ππ decays obtained within the dual approach to QCD with the one obtained recently by the RBC-UKQCD lattice approach to QCD and commented briefly on the status of the parameter BK and the ∆I = 1/2 rule
Summary
3.1 Preliminaries We will consider the usual basis of operators contributing to K → ππ amplitudes [34], namely. On the other hand, comparing (3.13) with the value for this matrix element obtained by RBC-UKQCD collaboration in [21] one extracts [39] All these results are very weakly dependent on the renormalization scale. The lattice results in [20] and [21] exhibit colour suppression of the matrix elements of Q3, Q5 and Q7 operators relative to the ones of Q4, Q6 and Q8, respectively: Q3 Q4. These results are consistent with the large N approach. Q1 0, which simplify the phenomenological analysis of ε′/ε in [40]
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