Analysis of theΔI=1/2rule andε′/εwith overlap fermions

We study the renormalization of the $\Delta S=1\mathrm{}$ effective weak Hamiltonian with overlap fermions. The mixing coefficients among dimension-6 operators are computed at one loop in perturbation theory. As a consequence of the chiral symmetry at finite lattice spacing and of the GIM mechanism, which is quadratic in the masses, the $\overrightarrow{K}\pi \pi$ and $\overrightarrow{K}\pi$ matrix elements relevant for the $\Delta I=1/2\mathrm{}$ rule can be computed without any power subtractions. The analogous amplitudes for ${\epsilon }^{\prime }/\epsilon$ require one divergent subtraction only, which can be performed nonperturbatively using $\overrightarrow{K}0$ matrix elements.