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The process $e^- e^+ \to \tau^- \tau^+$ is of particular interest because thetau lepton polarisation can be reconstructed, allowing its chiral nature to beprobed. This note reports on a study of the reconstruction of the di-tau finalstate at ILC-500, its selection and the reduction of backgrounds, theidentification of the tau lepton's decay mode, and on the extraction of the tauleptons' polarisation. The performance of this analysis is studied in twomodels of the ILD detector, one larger (IDR-L) the other smaller (IDR-S), whichdiffer in the outer radius of the TPC and of the subdetectors beyond, and inthe magnetic field strength of the detector solenoid. We find that thehigh-mass tau-pair events in which at least one tau decays haronically can beselected with an efficiency of around 60%, with a remaining background fromnon-di-tau processes at the few-% level. Single-prong decay modes $\tau^\pm \to\pi^\pm \nu, \tau^\pm \to \pi^\pm \pi^0 \nu, \tau^\pm \to \pi^\pm \pi^0 \pi^0\nu$ can be correctly identified in around 60-90% of cases, with samplepurities in the range 50-90%, depending on decay mode. The sensitivity to taupolarisation was estimated in the four beam polarisation datasets envisaged forthe $4\ ab^{-1}$ of data forseen for ILC-500. Statistical precisions on thepolarisation in the different datasets are predicted to be between 0.5 and 2%.While some small performance differences between the two detector models areseen, they have very similar final sensitivity to the polarisation measurement.

ILD benchmark: a study of $e^- e^+ \to \tau^- \tau^+$ at 500 GeV