Abstract
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 $K\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ decay and neutral kaon mixing. Our results are then compared to perturbation theory.
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