Opening the Rome-Southampton window for operator mixing matrices

TL;DR

This paper introduces a non-perturbative method to compute operator mixing matrices across a wider energy range, improving the analysis of four-quark operators relevant to kaon decays and mixing.

Contribution

The authors develop a continuum step scaling approach using RI-MOM scheme with twisted boundary conditions to relax the Rome-Southampton window bounds.

Findings

Successfully computed non-perturbative running matrices for four-quark operators.

Compared non-perturbative results with perturbation theory, showing consistency.

Enhanced the analysis of operators relevant to kaon physics.

Abstract

We show that the running of operators which mix under renormalization can be computed fully non-perturbatively 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 non-perturbative running matrix of four-quark operators needed in K->pipi decay and neutral kaon mixing. Our results are then compared to perturbation theory.