Matrix elements relevant forΔI=1/2rule andɛ′/ɛ from lattice QCD with staggered fermions

We perform a study of matrix elements relevant for the $\Delta I=1/2\mathrm{}$ rule and the direct $\mathrm{CP}$ violation parameter ${\varepsilon }^{\prime }/\varepsilon$ from first principles by computer simulation in lattice QCD. We use staggered (Kogut-Susskind) fermions, and employ chiral perturbation theory to study $K^0\to \pi \pi$ decays. Having obtained a reasonable statistical accuracy, we observe an enhancement of the $\Delta I=1/2\mathrm{}$ amplitude, consistent with experiment within our large systematic errors. Finite volume and quenching effects have been studied and were found small compared to noise. The estimates of ${\varepsilon }^{\prime }/\varepsilon$ are hindered by large uncertainties associated with operator matching. In this paper we explain the simulation method, present the results and address the systematic uncertainties.