B-physics through three-flavour lattice QCD with Wilson quarks

In heavy flavour phenomenology, the B-meson sector continues
to be a promising realm for precision tests of the Standard Model,
where evidence of New Physics can be expected. Lattice QCD offers
a powerful ab initio framework to reliably calculate the low-energy
hadronic matrix elements involved and thus to significantly reduce
the uncertainties of the theory inputs to such tests.
This talk discusses approaches to lattice B-physics that are based
on non-perturbative Heavy Quark Effective Theory (HQET) and step
scaling strategies. In particular, I will report on implementing a
suitable finite-volume matching procedure, to fix the parameters of
the effective theory, by numerical simulations with three flavours
of O($a$) improved Wilson quarks and how these HQET parameters can
be related to large-volume computations on (2+1)-flavour ensembles
to extract physical quantities. As another example, a precision
determination of the $B^*B\pi$ coupling in the limit of static
heavy and chiral light quarks is presented.