Calculation of $K\to \pi \pi$ decay amplitudes with improved Wilson fermion action 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. Expanding on the earlier works by Bernard et al. and by Donini et al., we show that mixings with four-fermion operators with wrong chirality are absent even for the Wilson fermion action for the parity odd process in both channels due to CPS symmetry. Therefore, after subtraction of an effect from the lower dimensional operator, a calculation of the decay amplitudes is possible without complications from operators with wrong chirality, as for the case with chirally symmetric lattice actions. As a first step to verify the possibility of calculations with the Wilson fermion action, we consider the decay amplitudes at an unphysical quark mass $m_K\sim 2m_{\pi }$. 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 $a=0.091\mathrm{fm}$, $m_{\pi }=280\mathrm{MeV}$, and $m_K=580\mathrm{MeV}$ on a ${32}^3\times 64$ ($La=2.9\mathrm{fm}$) lattice. For the quark loops in the penguin and disconnected contributions in the $I=0$ channel, the combined hopping parameter expansion and truncated solver method work very well for variance reduction. We obtain, for the first time with a Wilson-type fermion action, that $\mathrm{Re}{A}_0=60(36)\times 10^{-8}\mathrm{GeV}$ and $\mathrm{Im}{A}_0=-67(56)\times 10^{-12}\mathrm{GeV}$ for a matching scale $q^{*}=1/a$. The dependence on the matching scale $q^{*}$ for these values is weak.

Figure 1

Quark contractions for the time correlation function for the $K\to \pi \pi$ process for the operator $Q_i$ ($i=1,2,\dots ,10$).

Figure 2

Quark contractions for the time correlation function for the $K\to 0$ process for the operator $Q_i$ ($i=1,2,\dots ,10$).

Figure 3

Quark contractions for the time correlation function for the $K\to \pi \pi$ process for the operator $P=\overline{s}{\gamma }_{5}d$.

Figure 4

Quark contractions for the time correlation function for $\pi \pi \to \pi \pi$.

Figure 5

Four types of contractions for the time correlation function for $\pi \pi \to \pi \pi$. Left panel shows those for the point-wall function $G_{PW}^I(t)$ in (78) and right for the wall-wall function $G_{WW}^I(t)$ in (79).

Figure 6

Effective mass of the time correlation function $G_{PW}^I(t)$ for $\pi \pi \to \pi \pi$ with the isospin $I=0$ and $I=2$. Twice of the effective mass for the pion is also plotted for a comparison.

Figure 7

Effect of the truncated solver method. Panels 7 show the contributions of the contraction III and IV to the time correlation functions for ${\overline{Q}}_2$ and ${\overline{Q}}_6$ at $t=9$. In each panel, the data at $x=0$ show the contribution obtained by the usual stochastic method (74) with $N_R=1$, i.e., the contribution given by setting the quark loop $Q(\mathbf{x},t;\mathbf{x},t)={\xi }_i^{*}(\mathbf{x},t)S_i(\mathbf{x},t)$ for $i=1$. The data at $x=1,2,\dots ,6$ correspond to the contributions given by setting $Q(\mathbf{x},t;\mathbf{x},t)={\xi }_i^{*}(\mathbf{x},t)S_i^T(\mathbf{x},t)$ for $x=i=1,2,\dots ,6$ ($N_R+N_T=1+5$) with $S_i^T(\mathbf{x},t)$ in (77). The data at $x=7$ are average of the data at $x=1,2,\dots ,6$.

Figure 8

Time correlation function for the operator $Q_2$ for the $\mathrm{\Delta }I=1/2$ $K\to \pi \pi$ process, $G_2^{I=0}(t)$ in (29). The time slices of the two pion and the $K$ meson are set at $t_{\pi }=0$ and $t_K=24$, while the operator $Q_i$ runs over the whole time extent. (a) Contributions of the contraction III for $Q_2$, ${\alpha }_2·P$ and ${\overline{Q}}_2=Q_2-{\alpha }_2·P$. (b) Contributions of the contraction IV for $Q_2$, ${\alpha }_2·P$ and ${\overline{Q}}_2=Q_2-{\alpha }_2·P$. (c) Contributions from each type of contractions for ${\overline{Q}}_2$. (d) Total correlation functions calculated with TSM and without TSM.

Figure 9

Time correlation function for the operator $Q_6$ for the $\mathrm{\Delta }I=1/2$ $K\to \pi \pi$ process, $G_6^{I=0}(t)$ in (29), following the same convention as in Fig. 8.

Figure 10

Effective matrix element of $M_1^{I=2}(t)$ in (97) in the lattice unit. The time slice of the two pion is set at $t_{\pi }=0$. The operator $Q_i(t)$ runs over the whole time extent. The matrix elements given with the $K$ meson at time slice $t_K=22,24,26$ are plotted.

Figure 11

Effective matrix elements of $M_{7,8}^{I=2}(t)$ in (97), following the same convention as in Fig. 10.

Figure 12

Effective matrix elements of $M_{2,6}^{I=0}(t)$ in (97), following the same convention as in Fig. 10.