Redux on “When is the top quark a parton?”

If a new heavy particle ϕ is produced in association with the top quark in a hadron collider, the production cross section exhibits a collinear singularity of the form logðmϕ=mtÞ, which can be resummed by introducing a top quark parton distribution function (PDF). We reassess the necessity of such resummation in the context of a high-energy pp collider. We find that the introduction of a top PDF typically has a small effect at ffiffiffi S p ∼ 100 TeV due to three factors: (1) αs at the scale μ ¼ mϕ, which is quite small when logðmϕ=mtÞ is large, (2) the Bjorken x ≪ 1 for mϕ ≲ 10 TeV, and (3) the kinematic region where logðmϕ=mtÞ ≫ 1 is suppressed by phase space. We consider the example of pp → tHþ at next-to-leading logarithm (NLL) order and show that, in terms of the total cross section, the effect of a top PDF is generically smaller than that of a bottom PDF in the associated production of bϕ. However, in the pT distribution of the charged Higgs, the NLL calculation using a top PDF is crucial to generate the pT distribution for pT ≲ mt.