Dear Newton, the question you raised was discussed some time ago.
The point is that, in the 2+2 scheme that you have mentioned a) the two close doublets explain the solar and atmospheric phenomenology b. however, the 3 ordinary neutrinos must have to have a big admixture with the sterile one, in one of the two doublets or both c. this situation is not favoured by data analysis: we do not have any indication of disappearance in solar or atmospheric neutrinos.
Just to be sure, when we consider the 3+1 scheme, we can have a minor admixture only with the sterile state. In order to account for the anomalies, however, some mixing should be there.
In either cases, the mixing with the sterile neutrino is enough to bring it into equilibrium during the big bang. We should find one more neutrino species N(eff)=4 and also a non negligible neutrino mass, about 1 eV, or more in the 2+2 case. Neither of these predictions is in good agreement with Planck findings.
Cheers!
Dear Prof. Vissani,
Thanks for your reply, which is very informative.
best regards,
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