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

In this paper a general method to measure the longitudinal polarization of $\tau$ lepton in the process $pp \rightarrow Z^{0} \rightarrow \tau\tau$ 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 $\tau \rightarrow 3\pi\nu$ 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.