Soft integrals and soft anomalous dimensions at N3LO 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 homogeneous transcendental weight include first nine terms expansion dimensional regulator $\epsilon$. discuss application our to deeply-inelastic scattering $e^+e^-$ annihilation processes. use these compute coefficient for Drell-Yan gluon-fusion Higgs boson production higher orders $\epsilon$ QCD limit where only partons produced on top colorless final state. Furthermore, we extract anomalous dimension inclusive threshold function $N$-Jettiness beam jet N$^4$LO