3 neutrino schemes [18,19,20,21,22,23]

The simplest scheme, which accounts for the Solar (S) and Atmospheric (A) neutrino results, is that there are just three light neutrinos, all active, and that the mass eigenstates $\nu_i$ have masses in a hierarchy, analogous to the quarks and charged leptons. In that case, the atmospheric and solar neutrino mass-squared differences are measures of the mass-squares of the two heavier states, so that $m_3 \sim (\Delta m^2_{\rm atm})^{1/2} \sim 0.03-0.1$ eV; $m_2 \sim (\Delta m^2_{\rm solar})^{1/2} \sim 0.003$ eV (for MSW) or $\sim 10^{-5}$ eV (vacuum oscillations), and $m_1 \ll m_2$. The weak eigenstate neutrinos $\nu_a =(\nu_e, \nu_\mu, \nu_\tau)$ are related to the mass eigenstates $\nu_i$ by a unitary transformation $\nu_a = U_{ai} \nu_i$. If one makes the simplest assumption (from the Superkamiokande, CHOOZ, and Palo Verde data), that the $\nu_e$ decouples entirely from the atmospheric neutrino oscillations, $U_{e3}=0$, (of course, one can relax this assumption somewhat) and ignores possible CP-violating phases, then

$$\left( \begin{array}{c} \nu_e \\ \nu_\mu \\ \nu_\tau \end{array}\right) = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & c_\alpha & -s_\alpha \\ 0 & s_\alpha & c_\alpha \end{array}\right)$$

$$\mbox{$\times$} \left( \begin{array}{ccc} c_\theta & -s_\theta & 0 \\ s_\theta &... ...y}\right) \left( \begin{array}{c} \nu_1 \\ \nu_2 \\ \nu_3 \end{array}\right),$$ (10)

where $\alpha$ and $\theta$ are mixing angles associated with the atmospheric and solar neutrino oscillations, respectively, and where $c_\alpha \equiv \cos \alpha$, $s_\alpha \equiv \sin \alpha$, and similarly for $c_\theta, s_\theta$.

For maximal atmospheric neutrino mixing, $\sin^2 2 \alpha \sim 1$, this implies $c_\alpha = s_\alpha = 1/\sqrt{2}$, so that

$$U = \left( \begin{array}{ccc} c_\theta & -s_\theta & 0 \\ ... ...c{c_\theta}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{array}\right).$$ (11)

For small $\theta$, this implies that $\nu_{3,2} \sim \nu_{+,-} \equiv (\nu_\tau \pm \nu_\mu)/\sqrt{2}$ participate in atmospheric oscillations, while the solar neutrinos are associated with a small additional mixing between $\nu_e$ and $\nu_-$. Another limit, suggested by the possibility of vacuum oscillations for the solar neutrinos, is $\sin^2 2 \theta \sim 1$, or $c_\theta = s_\theta = 1/\sqrt{2}$, yielding

$$U = \left( \begin{array}{ccc} \frac{1}{\sqrt{2}} & -\frac{... ...c{1}{2} & \frac{1}{2} & \frac{1}{\sqrt{2}} \end{array}\right),$$ (12)

which is referred to as bi-maximal mixing [24,25,26]. A number of authors have discussed this pattern and how it might be obtained from models, as well as how much freedom there is to relax the assumptions of maximal atmospheric and solar mixing (the data actually allow and ) or the complete decoupling of $\nu_e$ from the atmospheric neutrinos. Another popular pattern,

$$U = \left( \begin{array}{ccc} \frac{1}{\sqrt{2}} & -\frac{... ... & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}\right),$$ (13)

known as democratic mixing [26], yields maximal solar oscillations and near-maximal ($8/9$) atmospheric oscillations.

The atmospheric neutrino data only determine the magnitude of $\vert\Delta m^2_{32}\vert \equiv \vert m_3^2 - m_2^2\vert = \Delta m^2_{\rm atm}$, so there is a variant inverted hierarchy in which $m_3 < m_1 < m_2$, with $m_2^2 - m_1^2 = \Delta m^2_{\rm solar} \ll m_2^2$, i.e., there is a quasi-degeneracy⁵, and $m_2^2 - m_3^2 = \Delta m^2_{\rm atm}$.

In the hierarchical and inverted patterns, the masses are all too small to be relevant to mixed dark matter, in which one of the components of the dark matter is hot, i.e., massive neutrinos. However, the solar and atmospheric oscillations only determine the differences in mass squares, so there are variants on these scenarios in which the three mass eigenstates are nearly degenerate rather than hierarchical, with small splittings associated with $\Delta m^2_{\rm atm}$ and $\Delta m^2_{\rm solar}$. For the common mass $m_{\rm av}$ in the 1 - several eV range, the hot dark matter could be relevant on large scales. Such mixed dark matter (with another, larger, component of cold dark matter accounting for smaller structures) models [27] were once quite popular. However, recent evidence for dark energy (cosmological constant or quintessence) eliminates most of the motivation for considering degenerate models, although they are still not excluded. Another problem with both the degenerate and inverted schemes is that the near degeneracies are usually unstable with respect to radiative corrections [28].

If the neutrinos are Majorana they could also lead to neutrinoless double beta decay, $\beta \beta_{0\nu}$ [29]. Current limits imply an upper limit of

$$\langle m_{\nu_e} \rangle = \sum_i \eta_i U_{ei}^2 \vert m_i\... ...0pt ;\hfil ...$$ (14)

on the effective mass for a mixture of light Majorana mass eigenstates, where $\eta_i$ is the CP-parity (or phase) of $\nu_i$. There is an additional uncertainty on the right due to the nuclear matrix elements. (There is no constraint on Dirac neutrinos.) This constraint is very important for the degenerate scheme for $m_{\rm av}$ in the eV range. The combination of small $\langle m_{\nu_e} \rangle \ll m_{\rm av}$, maximal atmospheric mixing, and $U_{e3}=0$ would imply cancellations, so that $\eta_1 \eta_2 = -1$ and $c_\theta = s_\theta = 1/\sqrt{2}$, i.e., maximal solar mixing. However, there is room to relax all of these assumptions considerably. For the hierarchical and inverted schemes $\langle m_{\nu_e} \rangle$ is small compared to the current experimental limit. However, proposed future experiments could be sensitive to the inverted case.