Inclusive Higgs measurements at the LHC have limited resolution on the gluon
fusion loops, being unable to distinguish the long-distance contributions
mediated by the top quark from possible short-distance new physics effects.
Using an Effective Field Theory (EFT) approach we compare several proposed
methods to lift this degeneracy, including tth and boosted, off-shell and
double Higgs production, and perform detailed projections to the
High-Luminosity LHC and a future hadron collider. We also point out the
sensitivity of the off-shell channel to modifications of the top-Z
couplings. By means of a general analysis we show that the reach is
comparable to that of tree-level processes such as ttZ production. We also
assess the regime of validity of the EFT, performing an explicit comparison
for a simple extension of the Standard Model containing one vector-like
quark.