  The data taken in Run II at the LHC have started to probe Higgs boson
production at high transverse momentum. Future data will provide a large sample
of events with boosted Higgs boson topologies, allowing for a detailed
understanding of electroweak Higgs boson plus two-jet production, and in
particular the vector-boson fusion mode (VBF). We perform a detailed comparison
of precision calculations for Higgs boson production in this channel, with
particular emphasis on large Higgs boson transverse momenta, and on the jet
radius dependence of the cross section. We study fixed-order predictions at NLO
and NNLO QCD, and compare the results to NLO plus parton shower (NLOPS) matched
calculations. The impact of the NNLO corrections on the central predictions is
mild, with inclusive scale uncertainties of the order of a few percent, which
can increase with the imposition of kinematic cuts. We find good agreement
between the fixed-order and matched calculations in non-Sudakov regions, and
the various NLOPS predictions also agree well in the Sudakov regime. We analyze
backgrounds to VBF Higgs boson production stemming from associated production,
and from gluon-gluon fusion. At high Higgs boson transverse momenta, the
$\Delta y_{jj}$ and/or $m_{jj}$ cuts typically used to enhance the VBF signal
over background lead to a reduced efficiency. We examine this effect as a
function of the jet radius and using different definitions of the tagging jets.
QCD radiative corrections increase for all Higgs production modes with
increasing Higgs boson $p_T$, but the proportionately larger increase in the
gluon fusion channel results in a decrease of the gluon-gluon fusion background
to electroweak Higgs plus two jet production upon requiring exclusive two-jet
topologies. We study this effect in detail and contrast in particular a central
jet veto with a global jet multiplicity requirement.
