Lattice determination of the $K\mathbf{\to }(\pi \pi {)}_{I=2}$ decay amplitude $A_2$

We describe the computation of the amplitude $A_2$ for a kaon to decay into two pions with isospin $I=2$. The results presented in [T. Blum et al., Phys. Rev. Lett. 108, 141601 (2012)] from an analysis of 63 gluon configurations are updated to 146 configurations giving $\mathrm{Re}{A}_2=1.381(46{)}_{\mathrm{stat}}(258{)}_{\mathrm{syst}}{10}^{-8}\mathrm{GeV}$ and $\mathrm{Im}{A}_2=-6.54(46{)}_{\mathrm{stat}}(120{)}_{\mathrm{syst}}{10}^{-13}\mathrm{GeV}$. $\mathrm{Re}{A}_2$ is in good agreement with the experimental result, whereas the value of $\mathrm{Im}{A}_2$ was hitherto unknown. We are also working toward a direct computation of the $K\to (\pi \pi {)}_{I=0}$ amplitude $A_0$ but, within the Standard Model, our result for $\mathrm{Im}{A}_2$ can be combined with the experimental results for $\mathrm{Re}{A}_0$, $\mathrm{Re}{A}_2$ and ${\epsilon }^{\prime }/\epsilon$ to give $\mathrm{Im}{A}_0/\mathrm{Re}{A}_0=-1.61(28)\times {10}^{-4}$. Our result for $\mathrm{Im}{A}_2$ implies that the electroweak penguin (EWP) contribution to ${\epsilon }^{\prime }/\epsilon$ is $\mathrm{Re}({\epsilon }^{\prime }/\epsilon {)}_{\mathrm{EWP}}=-(6.25\pm {0.44}_{\mathrm{stat}}\pm {1.19}_{\mathrm{syst}})\times {10}^{-4}$.

Figure 1

An illustration of around-the-world effects in the $K\to \pi \pi$ correlation function. In the left-hand figure the two-pion operator at $t=64$ annihilates one pion and creates another, while the weak Hamiltonian annihilates the kaon and a pion and creates a pion. The right-hand diagram illustrates the $K\to \pi \pi$ transition whose matrix element we evaluate.

Figure 2

Effective mass plots for the (a) pion and (b) kaon. Results for $m_{\pi }$ and $m_K$ obtained from the fits of the correlation functions to Eq. (12) are shown as the horizontal lines in each plot.

Figure 3

Effective energy plot for a pion with momentum $p=\sqrt{2}\pi /L$. The horizontal line corresponds to the value of $E_{\pi }$ obtained from a fit to Eq. (13).

Figure 4

The ratios $C_{\pi \pi }(t)/(C_{\pi }(t){)}^2$ defined in Eq. (15) at $p=0$ (a) and at $p=\sqrt{2}\pi /L$ (b). The minimum seen in the left-hand panel around $t=52$ results from the different large-time behavior of the numerator and denominator. While the denominator decreases exponentially as $t$ increases from 0 to 64, the numerator contains a small $t$-independent constant (caused by one backward propagating pion) which lessens its decrease at large time. If examined for $0\le t\le 128$ the ratio shown in the left-hand panel is symmetrical about the point $t=64$.

Figure 5

The ratios defined in Eq. (17) for $p=\sqrt{2}\pi /L$. The two-pion source is at $t=0$ while the kaon source is at $t_K=24$. Panel (a) indicates the (27, 1) operator, (b) the (8, 8) operator, and (c) the (8, 8) mix operator. The dashed line shows the error on the fit.

Figure 6

Plots of the $I=2$ two-pion $s$-wave phase shift against momentum $k_{\pi }$. Our results at $p=0$ and $p=\sqrt{2}\pi /L$ are denoted by the red circles and the dashed curve is the phenomenological representation from Ref. [32]. The left-hand plot is a comparison of the calculated phase shift with experimental results [30, 31, 32]. The right-hand plot is a zoom into the small $k_{\pi }$ region, demonstrating the approximate linear behavior of the phenomenological curve in the region of $p=0$. The scattering length used in the straight (dotted) line is calculated using chiral perturbation theory [61].

Figure 7

Schematic diagram illustrating the process in Eq. (43). In the diagram the arrows refer both to the flow of the indicated flavor quantum number and also to the indicated momentum. The spinor $(\alpha ,\beta ,\gamma ,\delta )$ and color $(i,j,k,l)$ labels are also indicated.

Figure 8

The running of the step-scaling function divided by the LO or NLO perturbative expression for the operator ${\mathcal{Q}}_{(27,1)}$ for the four RI-SMOM schemes considered in this paper. The ratio is set to 1 at $\mu =3\mathrm{GeV}$. The nonperturbative results have been extrapolated to both the chiral and continuum limits.

Figure 9

The running of the step-scaling function divided by the LO or NLO perturbative expression for the 77 element of the step-scaling function for the four RI-SMOM schemes considered in this paper. The ratio is set to 1 at $\mu =3\mathrm{GeV}$. The nonperturbative results have been extrapolated to both the chiral and continuum limits.

Figure 10

The running of the step-scaling function divided by the LO or NLO perturbative expression for the 88 element of the step-scaling function for the four RI-SMOM schemes considered in this paper. The ratio is set to 1 at $\mu =3\mathrm{GeV}$. The nonperturbative results have been extrapolated to both the chiral and continuum limits.

Figure 11

Reweighting $A_2$ from $m_l^{\mathrm{sea}}=0.001$ to $m_l^{\mathrm{sea}}=0.0001$. (a) Reweighting $\mathrm{Re}{A}_2$, (b) reweighting $\mathrm{Im}{A}_2$.