On the basis of a next-to-leading-order calculation in chiral perturbation theory, the first complete analysis of isospin breaking for direct $CP$ violation in $K^0\to 2\pi$ decays is performed. We find a destructive interference between three different sources of isospin violation in the $CP$ violation parameter ${ϵ}^{\prime }$. Within the uncertainties of large-$N_c$ estimates for the low-energy constants, the isospin violating correction for ${ϵ}^{\prime }$ is below 15%.

• Received 3 July 2003

DOI:https://doi.org/10.1103/PhysRevLett.91.162001