Momentum Dependence of the Penguin Interaction

We have considered the penguin interaction contribution to $K \rightarrow 2 \pi$ 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 $K \rightarrow \pi$. 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.