The Four Polarizations of the $W$ at High Energies

TL;DR

The paper develops a gauge-aware, diagrammatic framework to study polarization interference for intermediate $W$ bosons in high-energy processes, introducing an exact decomposition of polarization vectors via $oldsymbol{ ext{Θ}_{ ext{μν}}}$ and a reference vector $oldsymbol{n^ ext{μ}}$ to enable transparent mass-over-energy power counting. It shows that polarization interference does not vanish even near on-shell conditions, can be negative and significant in certain channels, and is generally suppressed at high energy due to helicity conservation, with distinct behavior across covariant, axial, and $R_\xi$ gauges. To stabilize gauge dependence, they propose the 2P scheme, merging longitudinal and scalar contributions into a single effective polarization, bringing covariant and axial gauges into closer agreement. The framework is applied to inclusive Drell–Yan, $W$+jets, top decays, and neutrino DIS, illustrating when polarization effects and interference are relevant or negligible for LHC-scale phenomenology. Overall, the work provides a scalable, gauge-consistent approach to polarized-vector-boson predictions and offers guidance for extending to multi-boson processes and potential new physics scenarios.

Abstract

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.

Figures (13)

• Figure 1: Graphical depiction of the matrix element for a resonant, unpolarized process $\mathcal{M}_{\rm unpol}^{\rm res}$, in terms of incoming/outgoing graphs $G_{in}^\mu$/$G_{out}^\nu$ and unpolarized propagator $\Pi^V_{\mu\nu}$, and its expansion in terms of polarized matrix elements and propagators $\mathcal{M}_{\lambda}$ and $\Pi^V_{\mu\nu}(\lambda)$.
• Figure 2: (L) Born-level diagram for the unpolarized, partonic process $q\bar{q}\rightarrow W^{(*)} \rightarrow f \bar{f}$ and its relationship to (R) the sum of helicity-polarized processes $q\bar{q}\rightarrow W^{(*)}_\lambda \rightarrow f \bar{f}$.
• Figure 3: Lowest order diagrams for the unpolarized partonic process $u\bar{d}\rightarrow W^{+(*)} g \rightarrow \nu_\tau \tau^+ g$, featuring a $(u\overline{d}g)$ current with (a) $d\to d^*g$ emission and a $t$-channel $d^*$ ($\mathcal{D}_{in}^\alpha$ in the main text), and (b) $u\to u^*g$ emission and a $t$-channel $u^*$ ($\mathcal{U}_{in}^\alpha$ in the main text).
• Figure 4: As a function of $W^{(*)}$ virtuality $\sqrt{q^2}$ and in the partonic center-of-mass frame at different phase space points, the squared matrix element for the unpolarized process $u\bar{d}\rightarrow W^{+(*)} g \rightarrow \nu_\tau \tau^+ g$ (solid), the squared amplitudes for transversely polarized $W_{\lambda=T}^{+(*)}$ (dash) and longitudinally polarized $W_{\lambda=T}^{+(*)}$ (dot), and the absolute value of the polarization interference (dash-dot).
• Figure 5: Upper: For (a) $\sqrt{s}=1{\rm\ TeV}$ and (b) $\sqrt{s}=13{\rm\ TeV}$, the hadronic cross sections for the unpolarized process $pp\rightarrow W^{\pm (*)} g \to \tau^\pm \nu$ (solid) as a function of the minimum gluon $p_T$, as well as the interference term $2|\vartheta|^2$ (dash), the interference term $2\Re[\mathcal{G}^*\vartheta]$ (dot), and the total interference (dash-dot). Middle and Lower: Ratio with respect to the unpolarized rate.