Measurement of the $\tau$ lepton polarization in Z boson decays in proton-proton collisions at $\sqrt{s}$ = 13 TeV

The polarization of $\tau$ leptons is measured using leptonic and hadronic $\tau$ lepton decays in Z $\to$ ${\tau }^{+}{\tau }^{-}$ events in proton-proton collisions at $\sqrt{s}$ = 13 TeV recorded by CMS at the CERN LHC with an integrated luminosity of 36.3 fb${}^{-1}$. The measured ${\tau }^{-}$ polarization at the Z boson mass pole is ${\mathcal{P}}_{\tau }$(Z) = $-$0.144 $\pm$ 0.006 (stat) $\pm$ 0.014 (syst) = $-$0.144 $\pm$ 0.015, in good agreement with the measurement of the $\tau$ lepton asymmetry parameter of $A_{\tau }$ = 0.1439 $\pm$ 0.0043 = $-{\mathcal{P}}_{\tau }$(Z) at LEP. The $\tau$ polarization depends on the ratio of the vector to axial-vector couplings of the $\tau$ leptons in the neutral current expression, and thus on the effective weak mixing angle ${\sin }^2{\theta }_{\mathrm{W}}^{\text{eff}}$, independently of the Z boson production mechanism. The obtained value ${\sin }^2{\theta }_{\mathrm{W}}^{\text{eff}}$ = 0.2319 $\pm$ 0.0008 (stat) $\pm$ 0.0018 (syst) = 0.2319 $\pm$ 0.0019 is in good agreement with measurements at e${}^{+}$e${}^{-}$ colliders.