In supersymmetry, the neutralino
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The mass splittings between the different neutralinos will dictate which patterns of decays are allowed.
Up to present, neutralinos have never been observed or detected in an experiment.
Origins in supersymmetric theories
In supersymmetry models, all Standard Model particles have partner particles with the same quantum numbers except for the quantum number spin, which differs by from its partner particle. Since the superpartners of the Z boson (zino), the photon (photino) and the neutral higgs (higgsino) have the same quantum numbers, they can mix to form four eigenstates of the mass operator called "neutralinos". In many models the lightest of the four neutralinos turns out to be the lightest supersymmetric particle (LSP), though other particles may also take on this role.
Phenomenology
The exact properties of each neutralino will depend on the details of the mixing (e.g. whether they are more higgsino-like or gaugino-like), but they tend to have masses at the weak scale (100 GeV ~ 1 TeV) and couple to other particles with strengths characteristic of the weak interaction. In this way, except for mass, they are phenomenologically similar to neutrinos, and so are not directly observable in particle detectors at accelerators.
In models in which R-parity is conserved and the lightest of the four neutralinos is the LSP, the lightest neutralino is stable and is eventually produced in the decay chain of all other superpartners.