Abstract
We present our result for the $K\to\pi\pi$ decay amplitudes for both the $\Delta I=1/2$ and $3/2$ processes with the improved Wilson fermion action. In order to realize the physical kinematics, where the pions in the final state have finite momenta, we consider the decay process $K({\bf p}) \to \pi({\bf p}) + \pi({\bf 0})$ in the nonzero momentum frame with momentum ${\bf p}=(0,0,2\pi/L)$ on the lattice. Our calculations are carried out with $N_f=2+1$ gauge configurations generated with the Iwasaki gauge action and nonperturbatively $O(a)$-improved Wilson fermion action at $a=0.091\,{\rm fm}$ ($1/a=2.176\,{\rm GeV}$), $m_\pi=260\,{\rm MeV}$, and $m_K=570\,{\rm MeV}$ on a $48^3\times 64$ ($La=4.4\,{\rm fm}$) lattice. For these parameters the energy of the $K$ meson is set at that of two-pion in the final state. We obtain ${\rm Re}A_2 = 2.431(19) \times10^{ -8}\,{\rm GeV}$, ${\rm Re}A_0 = 51(28) \times10^{ -8}\,{\rm GeV}$, and $\epsilon'/\epsilon = 1.9(5.7) \times10^{-3}$ for a matching scale $q^* =1/a$ where the errors are statistical. The dependence on the matching scale $q^*$ of these values is weak. The systematic error arising from the renormalization factors is expected to be around $1.3\%$ for ${\rm Re}A_2$ and $11 \%$ for ${\rm Re}A_0$. Prospects toward calculations with the physical quark mass are discussed.
Highlights
Calculation of the K → ππ decay amplitudes is very important to quantitatively understand the ΔI 1⁄4 1=2 rule of the K meson decay and to theoretically predict the direct CP violation parameter (ε0=ε) from the standard model
Expanding on the earlier works by Bernard et al [5] and by Donini et al [6], we showed in Ref. [7] that mixings with four-fermion operators with wrong chirality are absent even for the Wilson fermion action for the parity odd process in both the ΔI 1⁄4 3=2 and 1=2 channels due to CPS symmetry
As found in the previous work, we find that the first term of truncated solver method (TSM) in (77) in Ref. [7] is negligible for all time correlation functions in the present work, when the stopping condition is set at R < 1.2 × 10−6
Summary
Calculation of the K → ππ decay amplitudes is very important to quantitatively understand the ΔI 1⁄4 1=2 rule of the K meson decay and to theoretically predict the direct CP violation parameter (ε0=ε) from the standard model. The RBC-UKQCD Collaboration presented the results at the physical quark mass in Ref. In both the RBC-UKQCD and our calculations of the ΔI 1⁄4 1=2 process referred above, the kinematics was a K meson at rest decaying to two zero momentum pions at an unphysical quark mass satisfying mK ∼ 2mπ. The RBCUKQCD Collaboration reported a first direct calculation in the physical kinematics, where the pions in the final state have finite momenta, at the physical quark mass in Ref. IV conclusions of the present work are given and prospects toward calculations with the physical quark mass are discussed
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