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Preprint

Tau-lepton decays with up to two \(\pi^0\)'s in the final state, \(\tau^+ \to \pi^+ \bar{\nu}_\tau\),
\(\rho^+ (\pi^+\pi^0) \bar{\nu}_\tau\), \(a^+_1 (\pi^+\pi^0\pi^0) \bar{\nu}_\tau\), are
used to study the performance of the barrel part of the silicon-tungsten electromagnetic
calorimeter (Si-W ECAL) of the International Large Detector (ILD) at the future \(e^+-e^-\)
International Linear Collider. A correct reconstruction of the tau decay mode is crucial
for constraining the tau spin state and measuring the Higgs boson CP state in \(H\to
\tau^+\tau^-\) decays. About 95% of \(\pi^+ \bar{\nu}_\tau\) and 90% of \(\rho^+\bar{\nu}_\tau\)
and \(a^+_1\bar{\nu}_\tau\) decays from \(e^+e^-\to Z^0\to \tau^+\tau^-\) reaction at
\(e^\pm\)-beam energy of 125 GeV are correctly reconstructed. In a smaller ILD detector,
with Si-W ECAL radius reduced by about 20% these numbers degrade by at most 2%. The
\(\pi^0\) mass resolution stays below 10%. Since the failures in the tau-lepton reconstruction
are mainly due to photons, the increase of the ILD magnetic field from 3.5 T to 4
T does not bring any significant improvement.

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