The Four Polarizations of the $W$ at High Energies

We investigate polarization-induced interference and off-shell effects in predictions for high-energy, multi-leg processes with intermediate weak bosons carrying fixed helicities. Building on the ``truncated propagator’’ paradigm, we carry out our analysis at the level of helicity amplitudes and squared amplitudes. We introduce bookkeeping devices, suitable for covariant and axial gauge choices, that simplify the analytical evaluation of polarized amplitudes, and make power counting of mass-over-energy factors more manifest. Among other results, we show that polarization interference (i) is generally non-zero, even in on-shell limits, (ii) can be negative and comparable to longitudinal contributions, and (iii) is generated by helicity inversion and therefore suppressed (or zero) in high-energy limits for $s$- and $t$-channel exchanges. Connections between gauge invariance and the scalar polarization are also discussed, as is a scheme for reducing gauge dependence in predictions for polarized scattering rates. As case studies, we consider charged-current processes, including $W$(+jets), top quark decay, and neutrino deep-inelastic scattering.

💡 Research Summary

This paper presents a comprehensive analysis of polarization-induced interference and off-shell effects in high-energy, multi-leg scattering processes involving intermediate weak bosons, specifically W bosons, with fixed helicities. Motivated by the precision polarization measurement program at the LHC, the work critically examines the common assumption in the “template method” that interference between different helicity states of a resonant boson is negligible.

The authors build upon the “truncated propagator” paradigm, performing their analysis at the level of helicity amplitudes and squared amplitudes. They introduce novel bookkeeping devices, particularly a tensor decomposition of the polarization vector sum (Θμν), which simplifies the analytical evaluation of polarized amplitudes in covariant (Rξ and Unitary) and axial gauges. This formalism makes the power counting of mass-over-energy suppression factors more transparent and explicitly isolates terms responsible for helicity inversion.

The core theoretical findings are threefold: (1) Polarization interference (I_pol) is generally non-zero, persisting even in the on-shell limit of the weak boson. (2) This interference can be negative and reach magnitudes comparable to the contributions from longitudinally polarized states. (3) I_pol is generated by helicity-flipping amplitudes and is therefore suppressed or vanishes in the high-energy limit for s-channel and t-channel exchanges, explaining why neglecting it can be a reasonable approximation in certain kinematic regimes but not in general.

The paper deeply explores the connection between gauge invariance and the scalar polarization (λ=S), highlighting its role in canceling gauge dependencies among diagrams. To address the inherent gauge dependence of predictions for polarized scattering rates, the authors propose a “2P” (Two Physical Polarizations) scheme. This scheme redistributes the contributions from the scalar polarization into the transverse and longitudinal polarizations in a gauge-invariant manner, yielding more stable predictions.

These theoretical developments are validated through detailed case studies of realistic LHC processes, including W boson production in association with jets, top quark decay, and neutrino deep-inelastic scattering. Numerical simulations, particularly for pp → W + jets, demonstrate that the act of fixing the helicity of an intermediate W boson can indeed induce significant negative interference, comparable in size to the longitudinal contribution. This underscores the need for caution when interpreting polarization measurements based on the template method.

In conclusion, the work establishes that while polarization interference is suppressed at high energies, it is a non-negligible effect in off-shell and general kinematic configurations. It provides essential theoretical tools and insights for accurately interpreting current and future precision measurements of weak boson polarization at colliders, paving the way for more robust analyses in the search for new physics.