Nonperturbative renormalization of the Delta-S=1 weak Hamiltonian including the G_1 operator

Under renormalization, physical operators can mix with operators which vanish by the equations of motion. Such operators cannot contribute to matrix elements between physical states, but they contribute to operator mixing in renormalization schemes which are defined at an off-shell momentum point, such as the popular regularization-invariant schemes. For the first time, we renormalize the lattice $\Delta S=1$ effective weak Hamiltonian taking into account the most important such operator, $G_1 \propto \overline s \gamma_\nu (1-\gamma_5) D_\mu G_{\mu\nu} d$. This removes an important systematic error in calculations of weak matrix elements on the lattice.