An obvious isospin breaking correction to epsilon' of kinematical origin

Abstract

Isospin breaking correction to the $\Delta I = {1/2}$ decay of $K\to\pi\pi$ generates a large enough contribution to $\epsilon'$ through an induced $\Delta I = {3/2}$ amplitude in $K_L$ decay. Aside from the $\pi$-$\eta$ and $\pi$-$\eta'$ mixing contributions, there is a correction of kinematical origin due to the final-state $\pi^{\pm}$-$\pi^0$ mass difference, which is unambiguously calculable from the low-energy off-shell behavior of the $K\to\pi\pi$ amplitude. This correction to $\epsilon'$ reduces the isospin breaking parameter $\Omega_{IB}$ by 0.06, which is nearly one half of the $\pi$-$\eta$ mixing effect computed with chiral Lagrangians to $O(p^2)$.