Research

The polarisation describes the spin state of the vector boson with respect to a chosen spin-quantisation axis. In the helicity basis, this axis is chosen to be the direction of motion of the boson, and the helicity is defined as $$h = \frac{\vec{S} \cdot \vec{p}}{\vert \vec{p} \vert}\,.$$ The polarisation states are labelled by the helicity eigenvalue $\lambda = -1,0,+1$. For a $\mathrm{W}$ boson decaying into a lepton and a neutrino, the fractions of the three states are read from the angle $\theta^*$ of the charged lepton in the boson rest frame: $$\frac{1}{\sigma_{\mathrm{W}^\pm\to\ell^\pm\mathrm{\nu}_\ell}}\frac{\mathrm{d} \sigma_{\mathrm{W}^\pm\to\ell^\pm\mathrm{\nu}_\ell}}{\mathrm{d} \cos\theta^*} = \frac{3}{8}\left[ f_\mathrm{L}(1\mp\cos\theta^*)^2 +f_\mathrm{R}(1\pm\cos\theta^*)^2 +2f_0 (1-\cos^2\theta^*) \right],$$ with $f_\mathrm{R} + f_0 + f_\mathrm{L} = 1$: $f_0$ is the longitudinal fraction and $f_\mathrm{T} \equiv f_\mathrm{R} + f_\mathrm{L}$ the transverse one.