The ∆i = 1/2 Rule and Ε ′ /ε in the Chiral Quark Model

I discuss the role of the ∆I = 1/2 selection rule in K → ππ decays for the theoretical calculations of ε/ε . Lacking reliable “first principle” calculations, phenomenological approaches may help in understanding correlations among different contributions and available experimental data. In particular, in the chiral quark model approach the same dynamics which underlies the ∆I = 1/2 selection rule in kaon decays appears to enhance theK → ππ matrix elements of the gluonic penguins, thus driving ε/ε in the range of the recent experimental measurements. The results announced this year by the KTeV 1) and NA48 2) collaborations have marked a great experimental achievement, establishing 35 years -1 0 1 2 3 4 ¶ ¢¶ ́ 10 Muenchen H1996L Roma H1997L Trieste H1997L VSA HVNDR Figure 1: The combined 1-σ average of the NA31, E731, KTeV and NA48 results (ε/ε = 21.2±4.6×10−4) is shown by the gray horizontal band (the error is inflated according to the Particle Data Group procedure when averaging over data with substantially different central values). The old München, Roma and Trieste theoretical predictions for ε/ε are depicted by the vertical bars with their central values. For comparison, the VSA estimate is shown using two renormalization schemes. after the discovery of CP violation in the neutral kaon system 3) the existence of a much smaller violation acting directly in the decays. While the Standard Model (SM) of strong and electroweak interactions provides an economical and elegant understanding of indirect (ε) and direct (ε) CP violation in term of a single phase, the detailed calculation of the size of these effects implies mastering strong interactions at a scale where perturbative methods break down. In addition, CP violation in K → ππ decays is the result of a destructive interference between two sets of contributions, which may inflate up to an order of magnitude the uncertainties on the individual hadronic matrix elements of the effective four-quark operators. THis makes predicting ε/ε a complex and subtle theoretical challenge 4). In Fig. 1 I summarize the comparison of the theoretical predictions available before the KTeV announcement early this year with the present experimental data. The gray horizontal band shows the one-sigma average of the old NA31 (CERN) and E731 (Fermilab) data and the new KTeV and NA48 results. The vertical lines show the ranges of the published theoretical predictions (before February 1999), identified with the cities where most members of the groups reside. The range of the naive Vacuum Saturation Approximation (VSA) is shown for comparison. By considering the complexity of the problem, the theoretical calculations reported in Fig. 1, show a remarkable agreement, all of them pointing to a nonvanishing positive effect in the SM. On the other hand, if we focus our attention on the central values, the München (phenomenological 1/N) and Rome (lattice) calculations definitely prefer the 10 regime, contrary to the Trieste result which is above 10. Without entering the details of the calculations, it is important to emphasize that the abovementioned difference is mainly due to the different size of the hadronic matrix element of the gluonic penguin Q6 obtained in the various approaches. While the München and Rome calculations assume for 〈Q6〉 values in the neighboroud of the leading 1/N result (naive factorization), the Trieste calculation, based on the effective Chiral Quark Model (χQM) 5) and chiral expansion, finds a substantial enhancement of the I = 0 K → ππ amplitudes, which affect both current-current and penguin operators. The bulk of such an enhancement can be simply understood in terms of chiral dynamics (final-state interactions) relating the ε/ε prediction to the phenomenological embedding of the ∆I = 1/2 selection rule. The ∆I = 1/2 selection rule in K → ππ decays is known by some 45 years 6) and it states the experimental evidence that kaons are 400 times more likely to decay in the I = 0 two-pion state than in the I = 2 component. This rule is not justified by any general symmetry consideration and, although it is common understanding that its explanation must be rooted in the dynamics of strong interactions, there is no up to date derivation of this effect from first principle QCD. As summarized by Martinelli at this conference 7) lattice cannot provide us at present with reliable calculations of the I = 0 penguin operators relevant to ε/ε , as well as of the I = 0 components of the hadronic matrix elements of the tree-level current-current operators (penguin contractions), which are relevant for the ∆I = 1/2 selection rule. In the Münich approach 8) the ∆I = 1/2 rule is used in order to determine phenomenologically the matrix elements of Q1,2 and, via operatorial relations, some of the matrix elements of the left-handed penguins. Unfortunately, the approach does not allow for a phenomenological determination of the matrix elements of the penguin operators which are most relevant for ε/ε , namely the gluonic penguin Q6 and the electroweak penguin Q8. In the χQM approach, the hadronic matrix elements can be computed as an expansion in the external momenta in terms of three parameters: the constituent quark mass, the quark condensate and the gluon condensate. The Trieste group has computed the K → ππ matrix elements of the ∆S = 1, 2 effective lagrangian up to O(p) in the chiral expansion 9, 10). Hadronic matrix elements and short distance Wilson coefficients are then matched at a scale of 0.8 GeV as a reasonable compromise between the ranges of validity of perturbation theory and chiral lagrangian. By requiring the ∆I = 1/2 rule to be reproduced within a 20% uncertainty one obtains a phenomenological determination of the three basic parameters of the model. This step is crucial in order to make the model predictive, since there is no a-priori argument for the consistency of the matching procedure. As a matter of fact, all computed observables turn out to be very weakly scale (and renormalization scheme) dependent in a few hundred MeV range around the matching scale. Fig. 2 shows an anatomy of the various contributions which finally lead to the experimental value of the ∆I = 1/2 selection rule. Point (1) represents the result obtained by neglecting QCD and taking the factorized matrix element for the tree-level operator Q2, which is the leading electroweak contribution. The ratio A0/A2 is found equal to √ 2: by far off the experimental point (8). Step (2) includes the effects of perturbative QCD renormalization on the operatorsQ1,2 11). Step (3) shows the effect of including the gluonic penguin operators 12). Electroweak penguins 13) are numerically negligeable in the CP conserving amplitudes and are responsible for the very small shift in the A2 direction. Perturbative QCD and factorization lead us from (1) to (4). Non-factorizable gluon-condensate corrections, a crucial model dependent effect entering at the leading order in the chiral expansion, produce a substantial reduction of the A2 amplitude (5), as it was first observed by Pich and de Rafael 14). Moving the analysis to O(p) the chiral loop corrections, computed on the LO chiral lagrangian via dimensional regularization and minimal subtraction, lead us from (5) to (6), while the finite parts of the NLO counterterms 0.5 1 1.5 2 2.5 3 3.5 A(I=0) x 10^7 (GeV) 1 1.5 2 2.5 3 3.5 A (I = 2) x 1 0^ 8 ( G eV ) (1)