  Fixed-order perturbative calculations of fiducial cross sections for two-body
decay processes at colliders show disturbing sensitivity to unphysically low
momentum scales and, in the case of $H\to \gamma \gamma$ in gluon fusion, poor
convergence. Such problems have their origins in an interplay between the
behaviour of standard experimental cuts at small transverse momenta ($p_t$) and
logarithmic perturbative contributions. We illustrate how this interplay leads
to a factorially divergent structure in the perturbative series that sets in
already from the first orders. We propose simple modifications of fiducial cuts
to eliminate their key incriminating characteristic, a linear dependence of the
acceptance on the Higgs or $Z$-boson $p_t$, replacing it with quadratic
dependence. This brings major improvements in the behaviour of the perturbative
expansion. More elaborate cuts can achieve an acceptance that is independent of
the Higgs $p_t$ at low $p_t$, with a variety of consequent advantages.
