Perturbative renormalization of weak Hamiltonian four-fermion operators with overlap fermions

The renormalization of the most general dimension-six four-fermion operators without power subtractions is studied at one loop in lattice perturbation theory using overlap fermions. As expected, operators with different chirality do not mix among themselves and parity-conserving and parity-violating multiplets renormalize in the same way. The renormalization constants of unimproved and improved operators are also the same. These mixing factors are necessary to determine the physical matrix elements relevant to many phenomenological applications of weak interactions. The most important are the $K^0$-$K^0$ and $B^0$-$B^0$ mixings in the standard model and beyond, the $\Delta I=1/2\mathrm{}$ rule and ${\epsilon }^{\prime }/\epsilon .$