  A promising avenue to perform precision tests of the SM at the LHC is to
measure differential cross-sections at high invariant mass, exploiting in this
way the growth with the energy of the corrections induced by heavy new physics.
We classify the leading growing-with-energy effects in longitudinal diboson and
in associated Higgs production processes, showing that they can be encapsulated
in four real "high-energy primary" parameters. We assess the reach on these
parameters at the LHC and at future hadronic colliders, focusing in particular
on the fully leptonic $WZ$ channel that appears particularly promising. The
reach is found to be superior to existing constraints by one order of
magnitude, providing a test of the SM electroweak sector at the per-mille
level, in competition with LEP bounds. Unlike LHC Run-1 bounds, which only
apply to new physics effects that are much larger than the SM in the
high-energy tail of the distributions, the probe we study applies to a wider
class of new physics scenarios where such large departures are not expected.
