$K - \bar K$ mixing in a low energy effective QCD

Abstract

We consider the contribution from the siamese penguin diagram to$K - \bar K$ mixing. Combining some previous results, this contribution could be numerically significant, which is the motivation for the present calculation. For the siamese penguin, the effective lagrangian obtained from the perturbative region above ≈1 GeV is highly non-local. We account for such non-local effects by using an effective low-energy QCD, advocated by many authors, where the quarks are coupled to the pseudoscalar mesons π,K, η. Taking into account the momentum variation of the effective penguin interaction, it is found that the matrix element of the siamese penguin is finite in the limit$\Lambda _\chi \to \infty$, theCP-conserving part being about 50% of the well known box contribution. However, taking into account the finiteness of the cut-off, a numerical integration shows that the siamese penguin contribution is drastically reduced, and is only 0.2–0.3% of the box result.