We consider the contribution to K → 2π decays from the nondiagonal s → d quark transition amplitude. First, we calculate the most important part of the s → d transition, the so-called self-penguin amplitude ∼ G F α s , including the heavy top-quark case. Second, we calculate the matrix element of the s → d transition for the physical K → 2π process. This part of the analysis is performed within the Chiral Quark Model where quarks are coupled to the pseudoscalar mesons.The CP-conserving self-penguin contribution to K → 2π is found to be negligible. The obtained contribution to ǫ ′ /ǫ is sensitive to the values of the quark condensate and the constituent quark mass M. For reasonable values of these quantities we find that the self-penguin contribution to ǫ ′ /ǫ is 10-15 % of the gluonic penguin contribution and has the same sign. Given the large cancellation between gluonic and electroweak penguin contributions, this means that our contribution is of the same order of magnitude as ǫ ′ /ǫ itself.

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We have considered the penguin interaction contribution to K → 2π decays. In particular, we have investigated the effect of the momentum dependence of the penguin coefficient. Our analysis is performed within the Chiral Quark Model where quarks are coupled to the pseudoscalar mesons, which means that hadronic matrix elements can be calculated in terms of quark loop diagrams.We have inserted the momentum dependent penguin coefficient into the relevant quark loop diagrams for K → π. We discuss two possible prescriptions for performing the calculations, and conclude that the momentum dependence of the penguin coefficient increases the amplitude by 10-20 %. In any case, the (CP-conserving) penguin contribution is very sensitive to the values chosen for the involved parameters.

Within the Chiral Quark Model we have calculated the effect of non-factorizable G 3 contribution to the BK parameter. When all diagrams are summed, a vanishing result is obtained.