The rule and at O(p4) in the chiral expansion

Abstract

We calculate the hadronic matrix elements to O(p 4 ) in the chiral expansion for the (∆S = 1) K 0 → 2 π decays and the (∆S = 2)K 0 -K 0 oscillation. This is done within the framework of the chiral quark model. The chiral coefficients thus determined depend on the values of the quark and gluon condensates and the constituent quark mass. We show that it is possible to fit the ∆I = 1/2 rule of kaon decays with values of the condensates close to those obtained by QCD sum rules. The renormalization invariant amplitudes are obtained by matching the hadronic matrix elements and their chiral corrections to the short-distance NLO Wilson coefficients. For the same input values, we study the parameter B K of kaon oscillation and find B K = 1.1 ± 0.2. As an independent check, we determine B K from the experimental value of the K L -K S mass difference by using our calculation of the long-distance contributions. The destructive interplay between the short-and long-distance amplitudes yields B K = 1.2 ± 0.1, in agreement with the direct calculation.

Theoretical Estimate(s) of the CP-Violating Quantity ɛ′/ɛ in K → 2π decays

“…We find [17] that the ∆I = 1/2 rule is well reproduced for the reasonable values (see also [15]. Notice that δ G ≃ 2.…”

“…The hadronic matrix elements of Q i in (21) to O(p 4 ) has three terms: i) A three level term obtained from the O(p 2 ) chiral lagrangian (17).…”

“…For precice statements about numerical values, I refer to the cited papers. The strength of our approach [16,17,18] is that all contributions are calculated systematically up to O(p 4 ) in the chiral expansion. The weakness is its model dependence, which it shares more or less with other estimates.…”

Strong isospin-breaking effects in K→2π at next-to-leading order in the chiral expansion

“…QCD dressing, in the regime of scales > 1 GeV 2 , for which perturbative QCD can be sensibly employed, contribute a factor of 3 − 4 to the enhancement [1,2], while, in the case of non-leptonic K decay, long-distance effects, including those of final state interactions (FSI) [3][4][5] also contribute significantly. Attempts to provide a sensible matching of short and long distance effects in a single theoretical framework now appear likely to account for the full observed enhancement [3,[6][7][8]. The neglect of isospin breaking (IB), however, represents a potential problem for this putative understanding [9].…”

BK in the chiral limit within the 1/Nc expansion

“…We would like to insist on the fact that this result, contrary to the various large-N c inspired calculations which have been published so far [17,18,3,12,19,13,14] is the full result to next-to-leading order in the 1/N c expansion. The renormalization µ scale dependence as well as the scheme dependence in the factor S cancel exactly, at the nextto-leading log approximation, with the short distance µ and scheme dependences in the Wilson coefficient C ∆S=2 (µ) in Eq.…”