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. Working in the framework of the Effective $W$ Approximation (EWA) and in $R_\xi$ and axial gauges, we derive the unrenormalized, tree-level parton number density functions for weak bosons from massless leptons at next-to-leading power in the collinear expansion. Corrections exhibit a number of properties, including those conjectured but never universally derived. Parallels with heavy quark factorization are also found. We avoid pathologies through a novel set of kinematical consistency conditions. When satisfied, good agreement between the full and approximated matrix elements is achieved. Findings suggest that the EWA may be testable at the Large Hadron Collider with $450$ fb$^{-1}$ luminosity of same-sign $WW$ scattering data at $\sqrt{s}=13.6$ TeV.