The Four Polarizations of the $W$ at High Energies

Abstract

We investigate polarization-induced interference, off-shell effects, and gauge cancellations in predictions for high-energy, multi-leg processes with (near) resonant weak bosons. Building on the ``polarized propagator'' paradigm, we carry out our analysis at the level of helicity amplitudes and squared amplitudes, computing polarization interference directly. We introduce analytical decompositions of polarized propagators, valid for covariant and axial gauges, that simplify the organization and evaluation of polarized amplitudes, and make power counting of mass-over-energy factors manifest. We show: (i) For the fully massive case, polarization interference can exceed $\mathcal{O}(Γ_V^2/M_V^2)$ width-over-mass corrections, limiting the applicability of the narrow width and pole approximation at low energies. (ii) At the fully differential level, interference can naturally be larger than squared longitudinal amplitudes but can also vanish when bosons are emitted by unpolarized sources. (iii) When weak bosons decay to massless fermions, the non-interference of polarization after angular integration extends to the off-shell regime but remains approximate due to $V-A$ couplings. Guided by BRST invariance, we propose a simple scheme for grouping together polarizations that reduces gauge ambiguities in predictions for polarized scattering rates and is applicable to the fully massive case. As case studies, we examine polarization interference in $W$(+jets), top quark decay, and neutrino deep-inelastic scattering. For decays of unpolarized top quarks, interference exactly cancels at the totally unintegrated level.