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

Abstract

In this paper a general method to measure the longitudinal polarization of $τ$ lepton in the process $pp \rightarrow Z^{0} \rightarrow ττ$ is described. The method of optimal observable allows to distinguish between the $τ$ lepton helicity states with maximum possible sensitivity. The optimal observable for the main $τ$ decay modes ($e/μνν$, $πν$, $ρν$, $a_{1}ν$) that can be identified by CMS or ATLAS detectors is described. The analyzing power for $τ$ polarization can be further gained by taking into account a correlation between longitudinal spin states of $τ^{+}$ and $τ^{-}$. The theoretical systematic uncertainty on the $τ$ polarization measured using the $τ\rightarrow 3πν$ decays is estimated to be $(ΔP_τ)_{model} = (1.41 \pm 1.37)\times 10^{-4}$. This value is approximately 50 times smaller than the best uncertainty on $τ$ polarization measurement given by combination of all LEP results. An approach to propagate the measured $τ$ polarization averaged over the $Z^{0}$ lineshape and parton density functions to the electroweak parameters is also discussed.