Standard-model prediction for direct CP violation in $K\to \pi \pi$ decay

Abstract

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 $1/a=1.3784(68)$ 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$({\epsilon }^{\prime }/\epsilon )=1.38(5.15)(4.59)\times 10^{-4}$, 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 Re$({\epsilon }^{\prime }/\epsilon )$ provides a new test of the standard-model theory of CP violation, one which can be made more accurate with increasing computer capability.

Comments: 9 pages, 3 figures. Updated to match published version