On the perturbative renormalisation of four-quark operators for new physics

TL;DR

This paper analyzes the renormalization of four-quark operators relevant for beyond Standard Model physics, introducing Schrödinger Functional schemes and evaluating their perturbative properties and uncertainties in RG running.

Contribution

It introduces Schrödinger Functional schemes for non-perturbative renormalization of four-quark operators and computes their NLO anomalous dimensions in these schemes.

Findings

NLO anomalous dimensions determined for all $ abla F=1,2$ operators.

Large truncation effects observed in perturbative RG running.

Systematic uncertainties in NLO perturbation theory discussed.

Abstract

We discuss the renormalisation properties of the full set of $\Delta F=2$ operators involved in BSM processes, including the definition of RGI versions of operators that exhibit mixing under RG transformations. As a first step for a fully non-perturbative determination of the scale-dependent renormalization factors and their runnings, we introduce a family of appropriate Schr\"odinger Functional schemes, and study them in perturbation theory. This allows, in particular, to determine the NLO anomalous dimensions of all $\Delta F=1,2$ operators in these schemes. Finally, we discuss the systematic uncertainties related to the use of NLO perturbation theory for the RG running of four-quark operators to scales in the GeV range, in both our SF schemes and standard $\overline{MS}$ and RI-MOM schemes. Large truncation effects are found for some of the operators considered.