Isospin breaking corrections to a lattice QCD calculation of $\varepsilon'$

Abstract

Because of the $ΔI = 1/2$ rule, the effects of electromagnetism and the isospin-breaking light quark mass difference on the direct CP violation parameter $\varepsilon'$ may be as large as 25\% and are consequently of immediate interest. In a lattice QCD calculation the effects of isospin breaking on the various features of kaon decay can be clearly distinguished and those effects enhanced by the $ΔI=1/2$ rule on $\varepsilon'$ explicitly identified. We show that all such enhanced effects can be captured in a QCD + QED lattice calculation in which the exchanged photon has an energy in an accessible, intermediate range between 0.5-2.0 GeV. Short-distance effects ($2.0 \mathrm{\ GeV} \lesssim E_γ$), usually treated in QCD and electroweak perturbation theory, are not enhanced by the $ΔI=1/2$ rule, beyond the well-understood contribution of the two electroweak penguin operators. Infrared photons do not contribute to $\varepsilon'$ while low-energy photons ($E_γ\lesssim 0.5$ GeV) are not $ΔI=1/2$ rule enhanced or are suppressed by one order in chiral perturbation theory (ChPT). An explicit ChPT estimate of this low-energy-photon contribution, a contribution that is difficult to determine in a finite-volume lattice calculation, suggests that the effect on $\varepsilon'$ is on the order of 0.5\%.