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Experiments have not been able to detect any sign of neutrino's mass.
This only puts an upper limit on the mass, this limit however is very close to
zero for the electron neutrino. There are two other kinds of neutrinoes - mu and
tau. The upper limit on masses for these is fairly high.
Theory does not predict anything very strongly, though the standard model of
particle physics sets the masses of all three neutrinos to zero.
There are some problems like the solar neutrino problem and the atmospheric
neutrino problem, whose solution requires at least two neutrino species to be
massive. It also requires something called mixing of neutrino types (flavors). A
massive neutrino might aid formation of galaxies.
Lastly, there is a very strong upper limit on neutrino masses coming from
cosmology. This means that you cannot have massive neutrinos with an arbitrary
mass.
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