  We identify the differential cross sections for $t\bar t$ production and the
total cross section for Higgs production through gluon fusion as the processes
in which the two effective operators describing the leading non-standard
interactions of the top quark with the gluon can be disentangled and studied in
an independent fashion. Current data on the Higgs production and the $ {\rm
d}\sigma/{\rm d} {p^t_T}$ differential cross section provide limits comparable,
but not more stringent, than those from the total $t\bar t$ cross sections
measurements at the LHC and Tevatron, where however the two operators enter on
the same footing and can only be constrained together. Given the present
uncertainties, we find that the most stringent bounds are provided by a
combination of data on the $t \bar t$ total cross sections together with those
from the Higgs production. We conclude by stating the (modest) reduction in the
uncertainties necessary to provide more stringent limits by means of the Higgs
production and $t\bar t$ differential cross section observables at the LHC with
the future luminosity of 300 and 3000 fb$^{-1}$.
