The role of the threshold variable in soft-gluon resummation of the $t\bar{t}h$ production process

We study the role of the threshold variable in soft-gluon threshold resummation. We focus on the computation of the resummed total cross section, the final-state invariant-mass distribution, and transverse-momentum distribution of the Higgs boson when produced in association with a top-anti-top quark pair for the Large Hadron Collider operating at 13 TeV. We show that different choices for the threshold variable result in differences at next-to-leading power, i.e. contributions that are down by one power of the threshold variable. These contributions are noticeable numerically, although their effect on the resummed observables lies within the scale uncertainty of those observables. The average central results, obtained after combining several central-scale choices, agree remarkably well for different choices of the threshold variable. However, different threshold choices do effect the resulting scale uncertainty. To compute our results, we introduce a novel numerical method that we call the deformation method, which aids the stabilization of the inverse Mellin transform in cases where the analytical Mellin transform of the partonic cross section is unknown. We show that this method leads to a factor of 10 less function evaluations, while gaining a factor of 4-5 in numerical precision when compared to the standard method.