  Vector-boson scattering processes are of great importance for the current
run-II and future runs of the Large Hadron Collider. The presence of triple and
quartic gauge couplings in the process gives access to the gauge sector of the
Standard Model (SM) and possible new-physics contributions there. To test any
new-physics hypothesis, sound knowledge of the SM contributions is necessary,
with a precision which at least matches the experimental uncertainties of
existing and forthcoming measurements. In this article we present a detailed
study of the vector-boson scattering process with two positively-charged
leptons and missing transverse momentum in the final state. In particular, we
first carry out a systematic comparison of the various approximations that are
usually performed for this kind of process against the complete calculation, at
LO and NLO QCD accuracy. Such a study is performed both in the usual fiducial
region used by experimental collaborations and in a more inclusive phase space,
where the differences among the various approximations lead to more sizeable
effects. Afterwards, we turn to predictions matched to parton showers, at LO
and NLO: we show that on the one hand, the inclusion of NLO QCD corrections
leads to more stable predictions, but on the other hand the details of the
matching and of the parton-shower programs cause differences which are
considerably larger than those observed at fixed order, even in the
experimental fiducial region. We conclude with recommendations for experimental
studies of vector-boson scattering processes.
