No abstract

We study the CP -violating, |∆S| = 1 parameter ε ′ /ε by computing the hadronic matrix elements in the chiral quark model. We estimate in the chiral expansion the coefficients of the next-to-leading order terms that correspond to the operators Q 6 and Q 11 . We consider the impact of these corrections on the value of ε ′ /ε. We also investigate the possibility that the chiral quark model might drastically reduce the dimensional regularization γ 5 -scheme dependence of current evaluations of ε ′ /ε.

We present a new chiral quark model for mesons involving a heavy and a light (anti-) quark. The model relates various combinations of a quark -meson coupling constant and loop integrals to physical quantities. Then, some quantities may be predicted and some used as input. The extension from other similar models is that the present model includes the lowest order gluon condensate of the order (300 MeV) 4 determined by the mass splitting of the 0 − and the 1 − heavy meson states.Within the model, we find a reasonable description of parameters such as the decay constants f B and f D , the Isgur-Wise function and the axial vector coupling g A in chiral perturbation theory for light and heavy mesons.

After updating the determination of the combination of Kobayashi-Maskawa elements Im V td V * ts according to our new estimate of the parameter B K , we study the CP -violating ratio ε ′ /ε by means of hadronic matrix elements computed to O(p 4 ) in the chiral expansion. It is the first time that this order in chiral perturbation theory is included. We also discuss the relevance of some O(p 2 ) terms that are generally neglected in the calculation of the electroweak penguin matrix elements. The most important effect of this improved analysis is the substantial reduction (20%) of the leading electroweak penguin matrix element and accordingly a reduced cancelation between the electroweak and gluon penguin contributions. The ratio ε ′ /ε is thus larger than previously estimated and its predicted value enjoys a smaller uncertainty. We find ε ′ /ε = 1.7+1.4 −1.0 × 10 −3 . Values positive and of the order of 10 −3 are therefore preferred.

We discuss direct CP violation in the standard model by giving a new estimate of ε ′ /ε in kaon decays. Our analysis is based on the evaluation of the hadronic matrix elements of the ∆S = 1 effective quark lagrangian by means of the chiral quark model, with the inclusion of meson one-loop renormalization and NLO Wilson coefficients. Our estimate is fully consistent with the ∆I = 1/2 selection rule in K → ππ decays which is well reproduced within the same framework. By varying all parameters in the allowed ranges and, in particular, taking the quark condensate-which is the major source of uncertainty-between (−200 MeV) 3 and (−280 MeV) 3 we findAssuming for the quark condensate the improved PCAC result qq = −(221 ± 17 MeV) 3 and fixing ΛQCD to its central value, we find the more restrictive predictionwhere the central value is defined as the average over the allowed values of Im λ t in the first and second quadrants. In these estimates the relevant mixing parameter Im λ t is selfconsistently obtained from ε and we take m pole t = 180 ± 12 GeV. Our result is, to a very good approximation, renormalization-scale and γ 5 -scheme independent.

We use the chiral quark model to construct the complete O(p 2 ) weak ∆S = 1 chiral lagrangian via the bosonization of the ten relevant operators of the effective quark lagrangian. The chiral coefficients are given in terms of f π , the quark and gluon condensates and the scale-dependent NLO Wilson coefficients of the corresponding operators; in addition, they depend on the constituent quark mass M , a parameter characteristic of the model. All contributions of order N 2 c as well as N c and α s N c are included. The γ 5 -scheme dependence of the chiral coefficients, computed via dimensional regularization, and the Fierz transformation properties of the operator basis are discussed in detail. We apply our results to the evaluation of the hadronic matrix elements for the decays K → 2π, consistently including the renormalization induced by the meson loops. The effect of this renormalization is sizable and introduces a long-distance scale dependence that matches in the physical amplitudes the short-distance scale dependence of the Wilson coefficients.SISSA 43/95/EP September 1995 ‡ Permanent address:

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.

We study the B-parameter ("bag factor") for B − B mixing within a recently developed heavy-light chiral quark model. Non-factorizable contributions in terms of gluon condensates and chiral corrections are calculated. In addition, we also consider 1/m Q corrections within heavy quark effective field theory. Perturbative QCD effects below µ = m b known from other work are also included. Considering two sets of input parameters, we find that the renormalization invariant B-parameter isB = 1.51 ± 0.09 for B d andB = 1.40 ± 0.16 for B s .