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We study the contribution of $\nu_e \to \nu_\tau \to \tau \to \mu$transitions to the wrong-sign muon sample of the {\em golden} channel of theNeutrino Factory. Muons from tau decays are not really a background, since theycontain information from the oscillation signal, and represent a small fractionof the sample. However, if not properly handled they introduce serioussystematic error, in particular if the detector/analysis are sensitive to muonsof low energy. This systematic effect is particularly troublesome for large$\theta_{13} \geq 1^\circ$ and prevents the use of the Neutrino Factory as aprecision facility for large $\theta_{13}$. Such systematic error disappears ifthe tau contribution to the golden muon sample is taken into account. The factthat the fluxes of the Neutrino Factory are exactly calculable permits theknowledge of the tau sample due to the $\nu_e \to \nu_\tau$ oscillation. Wethen compute the contribution to the muon sample arising from this sample {\emin terms of the apparent muon energy}. This requires the computation of amigration matrix $M_{ij}$ which describes the contributions of the tauneutrinos of a given energy $E_i$, to the muon neutrinos of an apparent energy$E_j$. We demonstrate that applying $M_{ij}$ to the data permits the fullcorrection of the otherwise intolerable systematic error.

The $\tau$-contamination of the golden muon sample at the Neutrino Factory