Native transverse-momentum evolution and matching for TMD factorization and extraction

Abstract

We present a native transverse-momentum realization of the Collins/CSS2 leading-power transverse-momentum-dependent (TMD) factorization framework. The construction connects fixed-order distributional matching, origin contacts, unexpanded two-scale evolution, and a differentiable interface for hadronic structure. An endpoint-independent canonical scale with a specified infrared completion generates a Collins–Soper kernel on a common invariant domain. Complete canonical legs are bounded and admit constructive momentum-space evaluation through resolvent convolutions or elliptic solves. We give contact and finite-bin error bounds, parent-to-momentum-boundary translations, ordered flavor recurrences, and an all-order argument for the four exponent towers of $N^3\mathrm{LL}$ evolution. Localized joint-$(x,k_T)$ inputs, nonperturbative rapidity broadening, regulated moments and the full-spectrum subtraction are formulated as distinct parts of the prescription. The physical realization uses sourced four-loop beta/cusp and three-loop noncusp/Collins–Soper ingredients with strict-NLO hard/OPE boundaries. Non-Gaussian continuum tests and three-scale physical-PDF comparisons validate ordinary legs, contacts, individual insertions and analytic input derivatives. Including a declared bound on the continuation above a finite PDF table, physical bin discrepancies remain below $4.27\times10^{-5}$ of the peak reference bin. A qualified proton–proton Drell–Yan and neural-closure demonstration connects the method to inverse use. Higher-order boundary assembly and matched observables are specified without claiming an implemented $N^3\mathrm{LL}'$ spectrum or a completed global extraction.