Higher-order corrections for $tW$ production at high-energy hadron colliders

We discuss cross sections for $tW$ production in proton-proton collisions at the LHC and at higher-energy colliders with energies of up to 100 TeV. We find that, remarkably, the soft-gluon corrections are numerically dominant even at very high collider energies. We present results with soft-gluon corrections at approximate NNLO and approximate N$^3$LO matched to complete NLO results. These higher-order corrections are large and need to be included for better theoretical accuracy and smaller scale dependence. Total cross sections as well as top-quark and $W$-boson transverse-momentum and rapidity distributions are presented using various recent sets of parton distribution functions.