4 neutrino schemes [18,30,31,32]

The LSND results [33], if confirmed, would almost certainly imply a fourth, sterile, neutrino $\nu_s$ (the $Z$ lineshape does not allow a fourth light active neutrino), in which one or two of the neutrinos are separated from the others by $\sqrt{\Delta m^2_{\rm LSND}} \sim 0.4 - 1$ eV. There could be even more sterile neutrinos.

In the $3 + 1$ schemes, $\nu_4$ is heavier or lighter than $\nu_{1,2,3}$ by $\sqrt{\Delta m^2_{\rm LSND}}$, with the splittings between the latter controlled by $\Delta m^2_{\rm atm}$ and $\Delta m^2_{\rm solar}$. This case has generally been considered excluded by limits from $\nu_e$ and $\nu_\mu$ disappearance. However, small recent changes in the LSND favored range (to lower $\Delta m^2$) imply that these schemes are barely allowed for $\nu_4 \sim \nu_s$, and possibly for $\nu_4 \sim \nu_\tau$. The $\nu_4 \sim \nu_s$ case may offer a theoretical advantage over $2 + 2$ schemes in that the $\nu_s$ is more distinct from the active neutrinos.

In the $2 + 2$ schemes, one has two pairs of mass eigenstates $\nu_{1,2}$ and $\nu_{3,4}$, with $\pm \Delta m^2_{34} \sim \Delta m^2_{\rm atm} \sim 10^{-3}-10^{-2}$ eV$^2$, $\Delta m^2_{12} \sim \Delta m^2_{\rm solar} \sim 10^{-5}$ eV$^2$ (MSW) or $10^{-10}$ eV$^2$ (vacuum), and $\pm \Delta m^2_{24} \sim \Delta m^2_{\rm LSND} \sim 0.2-1$ eV$^2$, where $\Delta m^2_{ij} \equiv m_j^2 - m_i^2$. The reactor data imply that $\nu_e$ must be largely restricted to one of the pairs. The cases $\Delta m^2_{24} > 0$ and $<0$ are referred to as hierarchical and inverted, respectively. The inverted case, and to a somewhat lesser extent the hierarchical case, are quasi-degenerate, and may be unstable under radiative corrections [28].

The $2 + 2$ and some versions of the $3 + 1$ models involve a significant hot or warm neutrino component to the dark matter. The extra sterile neutrino may be of importance for big bang nucleosynthesis. The versions with $\nu_1$ in the heavier group may give significant contributions to $\beta \beta_{0\nu}$, although there may be major cancellations for large mixing.

The recent SuperKamiokande [19] and MACRO [34] atmospheric neutrino data exclude the pure $\nu_{\mu,s} \sim \nu_{3,4}$ case, in which the atmospheric neutrino results are associated with $\nu_\mu \mbox{$\rightarrow$}\nu_s$, while Super-K solar neutrino data [35] probably eliminates the pure $\nu_e \mbox{$\rightarrow$}\nu_s$ (i.e., $\nu_{e,s} \sim \nu_{1,2}$) explanation for the solar neutrinos. These were the simplest and perhaps most plausible cases. However, more general mixing schemes with significant $\nu_s$ admixtures in the solar and atmospheric neutrinos are possible [18,30,36].