There has been recent activity by astrophysicists to determine whether a fourth flavor of neutrino, a sterile neutrino, exists. It would likely be more massive than electron, muon or tau neutrinos. However, it wouldn't be affected by the weak force, only by gravity. It would therefore have similar characteristics to dark matter. The LHC is looking for evidence of supersymmetry, and the existence of the stable neutralino, another dark matter candidate. Can the LHC be of any assistance in helping determine the existence of the hypothetical sterile neutrino?
The LHC is a discovery machine. It was designed to maximize the probability of seeing new physics outside the Standard Model, but it is a poor experiment for discriminating between two or three states that the only signature they have is missing energy and small to zero missing mass.
The reason is that the strong interactions initiating the scattering are a many body problem, 3 quarks on three quarks and a great number of gluons on gluons, thus the only "closed" system is a subset defined by a high transverse momentum transfer, and this is contaminated by debris that is not really relevant to the deep scattering. The use of Monte Carlo simulations is imperative, and MC means that a model of the interactions is assumed in the creation of the sample. Not exactly circular, but not helping in fine details.
If the LHC finds photino or neutralino candidates a leptonic collider would be needed to be able to discriminate theories like the one proposing a fourth neutrino. IMO of course
The LHC is unlikely to be able to see the difference between a ordinary and a sterile neutrino. The LHC however can produce very accurate counts of interactions leading to missing energy, which would might be identified as neutrinos. The LHC will improve the accuracy of the Z invisible width, the number of times a Z boson decays into a neutrino. Zeds however are expected to decay only into ordinary neutrinos.
The term sterile ideally means that the particles not react with ordinary matter at all. Given this definition the LHC will never see a sterile neutrino.
Often the only interaction a sterile neutrino will undergo is an oscillation to and from an ordinary neutrino state. These would be unlikely to be seen at the LHC, instead you need a low energy neutrino beam with a near constant energy, a detector far away from the source, (like OPERA but without the varying energy of the input neutrinos). Sterile neutrino may already have shown up in nuclear reactor experiments where an anomaly in the count of neutrino measured seems to fit a sterile neutrino around the 1eV mark. 1 KeV sterile neutrino are considered a warm dark matter candidate, and also seem to help explain how supernova can exploded asymmetrically.