Weak bosons as partons below 10 TeV partonic center-of-momentum

We investigate the modeling of weak boson number densities for leptons and hadrons in practical calculations in the Standard Model. In the framework of the Effective $W$ Approximation (EWA) and in the $R_\xi$ and axial gauges, we derive the unrenormalized, tree-level parton number densities for weak bosons from massless fermions at next-to-leading power in the collinear expansion. Corrections exhibit various pathologies and properties, including those conjectured but not proven, and parallel heavy quark factorization. We suppress pathologies through a new set of kinematical consistency conditions. When satisfied, shapes and normalizations of full matrix elements for many-leg processes can be well approximated by the EWA and fragmentation contributions at leading power, suggesting the onset of tree-level factorization. Findings also suggest that the EWA is testable at the LHC with $\mathcal{L}=450$ fb$^{-1}$ of same-sign $WW$ scattering data at $\sqrt{s}=13.6$ TeV.