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Small-$x$ resummation has been proven recently to be a crucial ingredient fordescribing small-$x$ HERA data, and the inclusion of small-$x$ resummation inparton distribution function (PDF) determination has a sizeable effect on thePDFs even at the electroweak scale. In this work we explore the implications ofsmall-$x$ resummation at the Large Hadron Collider (LHC) and at a FutureCircular Collider (FCC). We construct the theoretical machinery for resummingphysical inclusive observables at hadron colliders, and describe itsimplementation in the public code HELL 3.0. We focus on Higgs production ingluon fusion as a prototypical example, both because it is sensitive tosmall-$x$ gluons and because of its importance for the LHC physics programme.We find that adding small-$x$ resummation to the N$^3$LO Higgs production crosssection can lead to an increase of up to 10% at FCC, while the effect issmaller (+1%) at LHC but still important to achieve a high level of precision.

Small-$x$ phenomenology at the LHC and beyond: HELL 3.0 and the case of the Higgs cross section