Measuring the polarization of boosted, hadronic $W$ bosons with jet substructure observables

In this work, we present a new technique to measure the longitudinal and transverse polarization fractions of hadronic decays of boosted $W$ bosons. We introduce a new jet substructure observable denoted as $p_\theta$, which is a proxy constructed purely out of subjet energies for the parton level decay polar angle of the $W$ boson in its rest-frame. The distribution of this observable is sensitive to the polarization of $W$ bosons and can therefore be used to reconstruct the $W$ polarization in a way that is independent of the production process -- assuming Standard Model (SM) rules governing decays. We argue that this proxy variable has lower reconstruction errors as compared to the other proxies that have been used by the experimental collaborations, especially for large boosts of the $W$-boson. As a test case, we study the efficacy of our technique on vector boson scattering (VBS) processes at the high luminosity Large Hadron Collider. We find that with only SM production channels, measuring the longitudinal polarization fraction is likely to be challenging even with 10 ab$^{-1}$ of data. We suggest further strategies and scenarios that may improve the prospects of measurement of the hadronic $W$ polarization fraction.