Calculation of Nonleptonic kaon decay amplitudes from Kπ matrix elements in quenched domain-wall QCD

We explore the application of the domain wall fermion formalism of lattice QCD to calculate the K ⃗ ππ decay amplitudes in terms of the K + →π + and K 0 →0 hadronic matrix elements through relations derived in chiral perturbation theory. Numerical simulations are carried out in quenched QCD using the domain-wall fermion action for quarks and a renormalization group-improved gauge action for gluons on a 16 3 ×32×16 and 24 3 ×32×16 lattice at β=2.6 corresponding to the lattice spacing 1/a≈2GeV. Quark loop contractions which appear in Penguin diagrams are calculated by the random noise method, and the ΔI=1/2 matrix elements which require subtractions with the quark loop contractions are obtained with a statistical accuracy of about 10%. We investigate the chiral properties required of the K + →π + matrix elements. Matching the lattice matrix elements to those in the continuum at μ=1/a using the perturbative renormalization factor to one loop order, and running to the scale μ=m c =1.3GeV with the renormalization group for N f =3 flavors, we calculate all the matrix elements needed for the decay amplitudes. With these matrix elements, the ΔI=3/2 decay amplitude ReA 2 shows a good agreement with experiment after an extrapolation to the chiral limit. The ΔI=1/2 amplitude ReA 0 , on the other hand, is about 50–60 % of the experimental one even after chiral extrapolation. In view of the insufficient enhancement of the ΔI=1/2 contribution, we employ the experimental values for the real parts of the decay amplitudes in our calculation of ɛ ′ /ɛ. The central values of our result indicate that the ΔI=3/2 contribution is larger than the ΔI=1/2 contribution so that ɛ ′ /ɛ is negative and has a magnitude of order 10 -4 . We discuss in detail possible systematic uncertainties, which seem too large for a definite conclusion on the value of ɛ ′ /ɛ