We compute the leading-order low-energy constants of the $\Delta S=1$ effective weak Hamiltonian in the quenched approximation of QCD with up, down, strange, and charm quarks degenerate and light. They are extracted by comparing the predictions of finite-volume chiral perturbation theory with lattice QCD computations of suitable correlation functions carried out with quark masses ranging from a few MeV up to half of the physical strange mass. We observe a $\Delta I=1/2$ enhancement in this corner of the parameter space of the theory. Although matching with the experimental result is not observed for the $\Delta I=1/2$ amplitude, our computation suggests large QCD contributions to the physical $\Delta I=1/2$ rule in the GIM limit, and represents the first step to quantify the role of the charm-quark mass in $K\to \pi \pi$ amplitudes. The use of fermions with an exact chiral symmetry is an essential ingredient in our computation.

• Received 2 August 2006

DOI:https://doi.org/10.1103/PhysRevLett.98.082003