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

TL;DR

This paper uses quenched domain-wall QCD to calculate $K o\pi o o$ decay amplitudes, comparing lattice results with experimental data and discussing discrepancies in the $ ext{I}=1/2$ channel.

Contribution

It introduces a method to compute $K o o o$ decay amplitudes from $K o o$ matrix elements using domain-wall fermions and quenched lattice QCD, with detailed renormalization and chiral extrapolation.

Findings

The $ ext{I}=3/2$ decay amplitude agrees with experiment in the chiral limit.

The $ ext{I}=1/2$ amplitude is only 50-60% of the experimental value.

The calculated $ ext{CP}$ violation parameter $ ext{Re}(rac{ ext{Epsilon}'}{ ext{Epsilon}})$ is negative and of order $10^{-4}$.

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 $\epsilon'/\epsilon$. We find that the $\Delta I=3/2$ contribution is larger than the $\Delta I=1/2$ contribution so that $\epsilon'/\epsilon$ is negative and has a magnitude of order $10^{-4}$. Possible reasons for these unsatisfactory results are discussed.