Constraints on top quark non-standard interactions from Higgs and $t \bar t$ production cross sections

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$^{-1}$.