Measurement of the tau lepton polarization in Z boson decays

The polarization of $\tau$ leptons is measured using leptonic and hadronic $\tau$ lepton decays in $\mathrm{Z}\rightarrow\tau\tau$ events. A sample of $\mathrm{pp}$ collisions of an integrated luminosity of $36.3\,\mathrm{fb}^{-1}$ at $\sqrt{s}=13\,\mathrm{TeV}$ is used. The measured polarization at the $\mathrm{Z}$ pole is $\mathcal{P}_{\tau}(\mathrm{Z}) = -0.144 \pm 0.015 = -0.144 \pm 0.006 \,(\text{stat}) \,\pm 0.014 \,(\text{syst})$, in good agreement with the standard model value of the lepton asymmetry parameter of $A_{\ell} = 0.1468 \pm 0.0003$. The $\tau$ polarization is used to determine the ratio of the vector to axial-vector coupling of the $\tau$ leptons in the neutral current and thus the weak mixing angle independently of the production process of the Z resonance. The obtained value is $ \sin^2\theta^{\mathrm{eff}}_W = 0.2319 \pm 0.0019 = 0.2319 \pm 0.0008\, (\text{stat}) \pm 0.0018\, (\text{syst})$.