Isospin breaking corrections to a lattice QCD calculation of ${\epsilon }^{\prime }$

Abstract

Because of the $\Delta I=1/2$ rule, the effects of electromagnetism and the isospin-breaking light quark mass difference on the direct CP violation parameter ${\epsilon }^{\prime }$ 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 $\Delta I=1/2$ rule on ${\epsilon }^{\prime }$ 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 ($2.0\mathrm{GeV}\lesssim E_{\gamma }$), usually treated in QCD and electroweak perturbation theory, are not enhanced by the $\Delta I=1/2$ rule, beyond the well-understood contribution of the two electroweak penguin operators. Infrared photons do not contribute to ${\epsilon }^{\prime }$ while low-energy photons ($E_{\gamma }\lesssim 0.5$ GeV) are not $\Delta I=1/2$ 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 ${\epsilon }^{\prime }$ is on the order of 0.5\%.

Comments: 26 pages, 2 figures