N$^3$LO soft-gluon corrections in single-particle-inclusive kinematics and $H^+ H^-$ production

We calculate the complete soft-gluon corrections for the production of colorless final states through N$^3$LO in single-particle-inclusive kinematics. We present explicit analytical results and use them to study higher-order QCD corrections for the production of a heavy charged Higgs pair ($H^+ H^-$) via quark-antiquark annihilation in the Two-Higgs-Doublet Model at LHC energies. We calculate the NNLO soft-gluon and virtual QCD corrections as well as the N$^3$LO soft-gluon corrections to the total cross section and the charged-Higgs rapidity distribution. This is the first calculation of complete N$^3$LO soft-gluon corrections for a process in single-particle-inclusive kinematics, and the results can be applied to other processes with colorless final states.