Methods for a measurement of $\tau$ polarization asymmetry in the decay $Z\to \tau \tau$ at LHC and determination of the effective weak mixing angle

Abstract

In this paper a general method to measure the longitudinal polarization of $\tau$ lepton in the process $pp\to Z^0\to \tau \tau$ is described. The method of optimal observable allows to distinguish between the $\tau$ lepton helicity states with maximum possible sensitivity. The optimal observable for the main $\tau$ decay modes ($e/\mu \nu \nu$, $\pi \nu$, $\rho \nu$, $a_1\nu$) that can be identified by CMS or ATLAS detectors is described. The analyzing power for $\tau$ polarization can be further gained by taking into account a correlation between longitudinal spin states of ${\tau }^{+}$ and ${\tau }^{-}$. The theoretical systematic uncertainty on the $\tau$ polarization measured using the $\tau \to 3\pi \nu$ decays is estimated to be $(\Delta P_{\tau }{)}_{model}=(1.41\pm 1.37)\times 10^{-4}$. This value is approximately 50 times smaller than the best uncertainty on $\tau$ polarization measurement given by combination of all LEP results. An approach to propagate the measured $\tau$ polarization averaged over the $Z^0$ lineshape and parton density functions to the electroweak parameters is also discussed.