$K\to (\pi \pi {)}_{I=2}$ Decay Amplitude from Lattice QCD

We report on the first realistic ab initio calculation of a hadronic weak decay, that of the amplitude $A_2$ for a kaon to decay into two $\pi$ mesons with isospin 2. We find $\mathrm{Re}{A}_2=(1.436\pm {0.063}_{\mathrm{stat}}\pm {0.258}_{\mathrm{syst}}){10}^{-8}\mathrm{GeV}$ in good agreement with the experimental result and for the hitherto unknown imaginary part we find $\mathrm{Im}{A}_2=-(6.83\pm {0.51}_{\mathrm{stat}}\pm {1.30}_{\mathrm{syst}}){10}^{-13}\mathrm{GeV}$. Moreover combining our result for $\mathrm{Im}{A}_2$ with experimental values of $\mathrm{Re}{A}_2$, $\mathrm{Re}{A}_0$, and ${ϵ}^{\prime }/ϵ$, we obtain the following value for the unknown ratio $\mathrm{Im}{A}_0/\mathrm{Re}{A}_0$ within the standard model: $\mathrm{Im}{A}_0/\mathrm{Re}{A}_0=-1.63(19{)}_{\mathrm{stat}}(20{)}_{\mathrm{syst}}\times {10}^{-4}$. One consequence of these results is that the contribution from $\mathrm{Im}{A}_2$ to the direct CP violation parameter ${ϵ}^{\prime }$ (the so-called Electroweak Penguin contribution) is $\mathrm{Re}({ϵ}^{\prime }/ϵ{)}_{\mathrm{EWP}}=-(6.52\pm {0.49}_{\mathrm{stat}}\pm {1.24}_{\mathrm{syst}})\times {10}^{-4}$. We explain why this calculation of $A_2$ represents a major milestone for lattice QCD and discuss the exciting prospects for a full quantitative understanding of $CP$ violation in kaon decays.

Figure 1

$R(t_Q)$ for the 3 operators which contribute to $K\to (\pi \pi {)}_{I=2}$ decay amplitudes: (a) $(\overline{s}d{)}_L(\overline{u}d{)}_L$, (b) $(\overline{s}d{)}_L(\overline{u}d{)}_R$, and (c) $({\overline{s}}^i{d}^j{)}_L({\overline{u}}^j{d}^i{)}_R$, where $(\overline{s}d{)}_L(\overline{u}d{)}_{L,R}=\mathbf{(}{\overline{s}}^i{\gamma }^{\mu }(1-{\gamma }^5)d^i\mathbf{)}\mathbf{(}{\overline{u}}^j{\gamma }_{\mu }(1\mp {\gamma }^5)d^j\mathbf{)}$. $i$, $j$ are color labels and $t_K$ and $t_{\pi \pi }$ are 0 and 24.