Rapidity distribution at soft-virtual and beyond for n-colorless particles to [Formula: see text] LO in QCD

We present a systematic framework to study the threshold contributions of the differential rapidity distribution for the production of any number of colorless particles in the hadronic colliders. This has been achieved based on the universality structure of the soft enhancements associated with the real emissions, along with the factorization property of the differential cross section and the renormalization group invariance. In this formalism, we present a universal soft-collinear operator to compute the soft virtual differential cross section for a generic [Formula: see text] scattering process up to next-to-next-to-next-to-next-to-leading order ([Formula: see text] LO) in perturbative QCD. We also provide a universal operator to perform the threshold resummation to next-to-next-to-next-to-leading logarithmic ([Formula: see text] LL) accuracy. We explicitly present the approximate analytical results of the rapidity distributions at [Formula: see text] LO and [Formula: see text] LL for the Higgs boson production through gluon fusion and bottom quark annihilation, and also for the Drell-Yan production at the hadronic collider. We extend our framework to include the next to threshold contributions for the diagonal partonic channels.