Neutrino physics is nowadays receiving more and more attention as a possible source of information for the long-standing problem of new physics beyond the Standard Model. The recent measurement of the mixing angle $theta_{13}$ in the standard mixing oscillation scenario encourages us to pursue the still missing results on leptonic CP violation and absolute neutrino masses. However, puzzling measurements exist that deserve an exhaustive evaluation. The NESSiE Collaboration has been setup to undertake conclusive experiments to clarify the muon-neutrino disappearance measurements at small $L/E$, which will be able to put severe constraints to models with more than the three-standard neutrinos, or even to robustly measure the presence of a new kind of neutrino oscillation for the first time. To this aim the use of the current FNAL-Booster neutrino beam for a Short-Baseline experiment has been carefully evaluated. This proposal refers to the use of magnetic spectrometers at two different sites, Near and Far. Their positions have been extensively studied, together with the possible performances of two OPERA-like spectrometers. The proposal is constrained by availability of existing hardware and a time-schedule compatible with the CERN project for a new more performant neutrino beam, which will nicely extend the physics results achievable at the Booster. The possible FNAL experiment will allow to clarify the current $ u_{mu}$ disappearance tension with $ u_e$ appearance and disappearance at the eV mass scale. Instead, a new CERN neutrino beam would allow a further span in the parameter space together with a refined control of systematics and, more relevant, the measurement of the antineutrino sector, by upgrading the spectrometer with detectors currently under R&D study.
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