We show that the running of operators which mix under renormalization can be computed fully nonperturbatively as a product of continuum step-scaling matrices. These step-scaling matrices are obtained by taking the “ratio” of $Z$ matrices computed at different energies in an RI-MOM type scheme for which twisted boundary conditions are an essential ingredient. Our method allows us to relax the bounds of the Rome-Southampton window. We also explain why such a method is important in view of the light quark physics program of the RBC-UKQCD Collaborations. To illustrate our method, using $n_f=2+1$ domain-wall fermions, we compute the nonperturbative running matrix of four-quark operators needed in $K\to \pi \pi$ decay and neutral kaon mixing. Our results are then compared to perturbation theory.

• Received 14 October 2011

DOI:https://doi.org/10.1103/PhysRevD.85.014501