The NESSiE Collaboration has been setup to undertake a conclusive experiment to clarify the {em muon--neutrino disappearance} measurements at short baselines in order to put severe constraints to models with more than the three--standard neutrinos. To this aim the current FNAL--Booster neutrino beam for a Short--Baseline experiment was carefully evaluated by considering the use of magnetic spectrometers at two sites, near and far ones. The detector locations were studied, together with the achievable performances of two OPERA--like spectrometers. The study was constrained by the availability of existing hardware and a time--schedule compatible with the undergoing project of multi--site Liquid--Argon detectors at FNAL. The settled physics case and the kind of proposed experiment on the Booster neutrino beam would definitively clarify the existing tension between the $ u_{mu}$ disappearance and the $ u_e$ appearance/disappearance at the eV mass scale. In the context of neutrino oscillations the measurement of $ u_{mu}$ disappearance is a robust and fast approach to either reject or discover new neutrino states at the eV mass scale. We discuss an experimental program able to extend by more than one order of magnitude (for neutrino disappearance) and by almost one order of magnitude (for antineutrino disappearance) the present range of sensitivity for the mixing angle between standard and sterile neutrinos. These extensions are larger than those achieved in any other proposal presented so far.
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