Soft Integrals and Soft Anomalous Dimensions at N$^3$LO and Beyond

We calculate soft phase-space and loop master integrals tor the computation of color-singlet cross sections through N$^3$LO in perturbative QCD. Our results are functions of homogeneous transcendental weight and include the first nine terms in the expansion in the dimensional regulator $\epsilon$. We discuss the application of our results to the computation of deeply-inelastic scattering and $e^+e^-$ annihilation processes. We use these results to compute the perturbative coefficient functions for the Drell-Yan and gluon-fusion Higgs boson production cross sections to higher orders in $\epsilon$ through N$^3$LO in QCD in the limit where only soft partons are produced on top of the colorless final state. Furthermore, we extract the anomalous dimension of the inclusive threshold soft function and of the $N$-Jettiness beam and jet functions to N$^4$LO in perturbative QCD.