Calculation of $K\to \pi \pi$ decay amplitudes with an improved Wilson fermion action in a nonzero momentum frame in lattice QCD

We present our result for the $K\to \pi \pi$ decay amplitudes for both the $\mathrm{\Delta }I=1/2$ 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 $K(\mathbf{p})\to \pi (\mathbf{p})+\pi (\mathbf{0})$ in the nonzero momentum frame with momentum $\mathbf{p}=(0,0,2\pi /L)$ 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 $a=0.091\mathrm{fm}$ ($1/a=2.176\mathrm{GeV}$), $m_{\pi }=260\mathrm{MeV}$, and $m_K=570\mathrm{MeV}$ on a $48^3\times 64$ ($La=4.4\mathrm{fm}$) lattice. For these parameters the energy of the $K$ meson is set at that of two-pion in the final state. We obtain $\mathrm{Re}{A}_2=2.431(19)\times {10}^{-8}\mathrm{GeV}$, $\mathrm{Re}{A}_0=51(28)\times {10}^{-8}\mathrm{GeV}$, and ${\epsilon }^{\prime }/\epsilon =1.9(5.7)\times {10}^{-3}$ 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 $\mathrm{Re}{A}_2$ and 11% for $\mathrm{Re}{A}_0$. Prospects toward calculations with the physical quark mass are discussed.

Figure 1

Effective energy of the time correlation function $G_{PW}^I(t,t_1,t_2)$ for the $\pi \pi \to \pi \pi$ with the isospin $I=0$ and $I=2$. The source operator for the two-pion is located at $(t_1,t_2)=(0,4)$. A value in the noninteraction case given by adding the effective energy of the pion with the momentum $\mathbf{p}=(0,0,2\pi /L)$ and that for the zero momentum, which corresponds to $E_{\mathrm{free}}=\sqrt{m_{\pi }^2+p^2}+m_{\pi }$, is plotted by blue symbol for a comparison.

Figure 2

Effective matrix element of $M_i^{I=2}(t)$ in (36) for the operator $Q_1$ in the lattice unit. The operator for the two-pion is located at $(t_1,t_2)=(0,4)$ and the $K$ meson at $t_K=29$. The operator $Q_i(t)$ runs over the whole time extent. The result of the fitting to (43) in the time range $t=[13,18]$ is plotted by a curved line and the result of the $K\to \pi \pi$ matrix element is shown by a straight line with one sigma band.

Figure 3

Effective matrix elements of $M_i^{I=2}(t)$ in (36) for the operators $Q_{7,8}$, following the same convention as in Fig. 2.

Figure 4

Effective matrix element of $M_i^{I=0}(t)$ in (36) for the LL-type four-fermion operators $Q_{1,2,3,4,9,10}$, following the same convention as in Fig. 2.

Figure 5

Effective matrix element of $M_i^{I=0}(t)$ in (36) for the LR-type four-fermion operators $Q_{5,6,7,8}$, following the same convention as in Fig. 2.