A strategy for implementing non-perturbative renormalisation of heavy-light four-quark operators in the static approximation

TL;DR

This paper develops a non-perturbative renormalisation strategy for heavy-light four-quark operators in the static approximation, using the Schrödinger functional and twisted-mass QCD to improve accuracy and symmetry considerations.

Contribution

It introduces a method combining Schrödinger functional techniques and twisted-mass QCD to non-perturbatively renormalise heavy-light four-quark operators, addressing mixing and symmetry constraints.

Findings

Verified mixing patterns at one-loop lattice perturbation theory

Determined anomalous dimensions at NLO in the Schrödinger functional scheme

Showed how to avoid finite subtractions from chiral symmetry breaking

Abstract

We discuss the renormalisation properties of the complete set of $\Delta B = 2$ four-quark operators with the heavy quark treated in the static approximation. We elucidate the role of heavy quark symmetry and other symmetry transformations in constraining their mixing under renormalisation. By employing the Schroedinger functional, a set of non-perturbative renormalisation conditions can be defined in terms of suitable correlation functions. As a first step in a fully non-perturbative determination of the scale-dependent renormalisation factors, we evaluate these conditions in lattice perturbation theory at one loop. Thereby we verify the expected mixing patterns and determine the anomalous dimensions of the operators at NLO in the Schroedinger functional scheme. Finally, by employing twisted-mass QCD it is shown how finite subtractions arising from explicit chiral symmetry breaking can be avoided completely.