Calculation of nonleptonic kaon decay amplitudes fromK→πmatrix elements in quenched domain-wall QCD

We explore the application of the domain wall fermion formalism of lattice QCD to calculate the $\overrightarrow{K}\pi \pi$ decay amplitudes in terms of the $K^{+}\to {\pi }^{+}$ and $K^0\to 0$ 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 ${16}^3\times 32\times 16$ and ${24}^3\times 32\times 16$ lattice at $\beta =2.6\mathrm{}$ corresponding to the lattice spacing $1/a\approx 2\mathrm{GeV}.$ Quark loop contractions which appear in Penguin diagrams are calculated by the random noise method, and the $\Delta I=1/2\mathrm{}$ 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 $K^{+}\to {\pi }^{+}$ matrix elements. Matching the lattice matrix elements to those in the continuum at $\mu =1/a$ using the perturbative renormalization factor to one loop order, and running to the scale $\mu {=m}_c=1.3\mathrm{}\mathrm{GeV}$ with the renormalization group for $N_f=3\mathrm{}$ flavors, we calculate all the matrix elements needed for the decay amplitudes. With these matrix elements, the $\Delta I=3/2\mathrm{}$ decay amplitude $\mathrm{Re}{A}_2$ shows a good agreement with experiment after an extrapolation to the chiral limit. The $\Delta I=1/2\mathrm{}$ amplitude $\mathrm{Re}{A}_0,$ on the other hand, is about 50–60 % of the experimental one even after chiral extrapolation. In view of the insufficient enhancement of the $\Delta I=1/2\mathrm{}$ contribution, we employ the experimental values for the real parts of the decay amplitudes in our calculation of ${\varepsilon }^{\prime }/\varepsilon .$ The central values of our result indicate that the $\Delta I=3/2\mathrm{}$ contribution is larger than the $\Delta I=1/2\mathrm{}$ contribution so that ${\varepsilon }^{\prime }/\varepsilon$ 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 ${\varepsilon }^{\prime }/\varepsilon .$