$N_f=2+1+1$ renormalisation of four-quark operators

TL;DR

This paper investigates the renormalisation of a set of four-quark operators relevant for lattice QCD, demonstrating that discretisation effects are manageable and allowing for high-precision future studies using Rome-Southampton renormalisation.

Contribution

The study extends previous $B_K$ renormalisation strategies to a broader set of operators, showing discretisation effects are largely independent of lattice spacing.

Findings

Discretisation effects are independent of lattice spacing.

Effects remain controlled up to high energy scales.

Varying the intermediate scale impacts perturbative matching prospects.

Abstract

When several four-quark operators are allowed to mix through renormalisation, this can considerably amplify the problems coming from perturbative truncation and discretisation effects. In this work we investigate whether our previous $B_K$ strategy can conveniently be generalised to a wider set of operators, corresponding to the so-called "SUSY $B_K$" basis of four-quark operators. We show that the discretisation effects, when plotted as a function of $ap$, are surprisingly independent of the lattice spacing. They appear reasonably under control up to very large energy scales. This allows us to discuss the effect of varying the intermediate scale on which the perturbative matching is done, and therefore the prospects of future high-precision studies with a Rome-Southampton renormalisation.