Calculation of Non-Leptonic Kaon Decay Amplitudes from $K\to \pi$ Matrix Elements in Quenched Domain-Wall QCD

Abstract

We explore application of the domain wall fermion formalism of lattice QCD to calculate the $K\to \pi \pi$ decay amplitudes in terms of the $K\to \pi$ and $K\to 0$ hadronic matrix elements through relations derived in chiral perturbation theory. Numerical simulations are carried out in quenched QCD using domain-wall fermion action for quarks and an RG-improved gauge action for gluons on a $16^3\times 32\times 16$ and $24^3\times 32\times 16$ lattice at $\beta =2.6$ corresponding to the lattice spacing $1/a\approx 2$GeV. Quark loop contractions which appear in Penguin diagrams are calculated by the random noise method, and the $\Delta I=1/2$ matrix elements which require subtractions with the quark loop contractions are obtained with a statistical accuracy of about 10%. We confirm 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 $\mu =m_c=1.3$ GeV 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 $\Delta I=3/2$ decay amplitude shows a good agreement with experiment in the chiral limit. The $\Delta I=1/2$ amplitude, on the other hand, is about 50--60% of the experimental one even after chiral extrapolation. In view ofthe insufficient enhancement of the $\Delta I=1/2$ contribution, we employ the experimental values for the real parts of the decay amplitudes in our calculation of ${\epsilon }^{\prime }/\epsilon$. We find that the $\Delta I=3/2$ contribution is larger than the $\Delta I=1/2$ contribution so that ${\epsilon }^{\prime }/\epsilon$ is negative and has a magnitude of order $10^{-4}$. Possible reasons for these unsatisfactory results are discussed.

arXiv comment48 pages, final version to appear in Physical Review D

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