  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.
