$W$ boson polarization in vector boson scattering at the LHC

TL;DR

This paper proposes a gauge-invariant method to define and analyze the polarization states of W bosons in vector boson scattering at the LHC, addressing conceptual and practical challenges in measuring polarization fractions.

Contribution

It introduces a new approach to define polarized W boson cross sections by dropping non-resonant diagrams and projecting on-shell, enabling polarization measurement in realistic LHC conditions.

Findings

The sum of polarized distributions reproduces full results.

Ratios of polarized to full cross sections match Legendre polynomial projections without cuts.

Angular distribution shapes can be used to extract polarization fractions with selection cuts.

Abstract

Measuring the scattering of longitudinally-polarized vector bosons represents a fundamental test of ElectroWeak Symmetry Breaking. In addition to the challenges provided by low rates and large backgrounds, there are conceptual issues which need to be clarified for the definition of a suitable signal. Since vector bosons are unstable and can only be observed through their decay products, the polarization states interfere among themselves. Moreover, already at tree level, there are diagrams which cannot be interpreted as production times decay of EW bosons but are necessary for gauge invariance. We discuss a possible way to define a cross section for polarized $W$'s, dropping all non resonant diagrams, and projecting on shell the resonant ones, thus, preserving gauge invariance. In most cases, the sum of polarized distributions reproduces the full results. In the absence of cuts, the ratios of the polarized cross sections to the full one agree with the results of a standard projection on Legendre polynomials. While the latter cannot be employed in a realistic environment, a comparison of the data with the shapes of the angular distributions for polarized vector bosons allows the extraction of the polarization fractions in the presence of selection cuts on the charged leptons.