Standard-model prediction for direct CP violation in $Ktopipi$ decay

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