Neutrinos are copiously produced at particle colliders, but no collider neutrino has ever been detected. Colliders, and particularly hadron colliders, produce both neutrinos and anti-neutrinos of all flavors at very high energies, and they are therefore highly complementary to those from other sources. FASER, the recently approved Forward Search Experiment at the Large Hadron Collider, is ideally located to provide the first detection and study of collider neutrinos. We investigate the prospects for neutrino studies of a proposed component of FASER, FASER$ u$, a 25cm x 25cm x 1.35m emulsion detector to be placed directly in front of the FASER spectrometer in tunnel TI12. FASER$ u$ consists of 1000 layers of emulsion films interleaved with 1-mm-thick tungsten plates, with a total tungsten target mass of 1.2 tons. We estimate the neutrino fluxes and interaction rates at FASER$ u$, describe the FASER$ u$ detector, and analyze the characteristics of the signals and primary backgrounds. For an integrated luminosity of 150 fb$^{-1}$ to be collected during Run 3 of the 14 TeV Large Hadron Collider from 2021-23, and assuming standard model cross sections, approximately 1300 electron neutrinos, 20,000 muon neutrinos, and 20 tau neutrinos will interact in FASER$ u$, with mean energies of 600 GeV to 1 TeV, depending on the flavor. With such rates and energies, FASER will measure neutrino cross sections at energies where they are currently unconstrained, will bound models of forward particle production, and could open a new window on physics beyond the standard model.
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