Standard Model Prediction for Direct $CP$ Violation in $K\to \pi \pi$ Decay

We report the first lattice QCD calculation of the complex kaon decay amplitude $A_0$ with physical kinematics, using a ${32}^3\times 64$ lattice volume and a single lattice spacing $a$, with $1/a=1.3784(68)\mathrm{GeV}$. We find $\mathrm{Re}(A_0)=4.66(1.00)(1.26)\times 10^{-7}\mathrm{GeV}$ and $\mathrm{Im}(A_0)=-1.90(1.23)(1.08)\times 10^{-11}\mathrm{GeV}$, where the first error is statistical and the second systematic. The first value is in approximate agreement with the experimental result: $\mathrm{Re}(A_0)=3.3201(18)\times 10^{-7}\mathrm{GeV}$, while the second can be used to compute the direct $CP$-violating ratio $\mathrm{Re}({ϵ}^{\prime }/ϵ)=1.38(5.15)(4.59)\times 10^{-4}$, which is $2.1\sigma$ below the experimental value $16.6(2.3)\times 10^{-4}$. The real part of $A_0$ is $CP$ conserving and serves as a test of our method while the result for $\mathrm{Re}({ϵ}^{\prime }/ϵ)$ provides a new test of the standard model theory of $CP$ violation, one which can be made more accurate with increasing computer capability.

Figure 1

Examples of the four types of diagram contributing to the $\mathrm{\Delta }I=1/2$, $K\to \pi \pi$ decay. Lines labeled $\ell$ or $s$ represent light or strange quarks. Unlabeled lines are light quarks.

Figure 2

Effective energies of the kaon (squares) and two-pion (circles) states deduced from the corresponding two-point functions by equating the results from two time separations to the function $A\cosh E_{\text{eff}}(T/2-t)$, where $T=64$ is the temporal lattice size, plotted as a function of the smallest of those two separations. (We replace $T$ by $T-8$ for the $\pi \pi$ case.) These are overlaid by the error bands corresponding to the fitted values of $E_{\pi \pi }$ (light blue) and $m_K$ (pink).

Figure 3

The $Q_2$ and $Q_6$ three-point functions, plotted in lattice units as functions of $t_{\pi \pi }-t_Q$, with the time dependence in Eq. (3) removed. The horizontal lines show the central value and errors from the fit described below.