The observation of the neutrino oscillations in experiments with atmospheric, solar, reactor and accelerator neutrinos proves that neutrino masses are different from zero and that the states of flavor neutrinos e, μ, tau are mixtures of states of neutrinos with different masses. There are two general possibilities for neutrinos with definite masses: they can be 4-component Dirac particles, possessing conserved total lepton number which distinguish neutrinos and antineutrinos or purely neutral 2-component Majorana particles with identical neutrinos and antineutrinos. . . .
Neutrino masses are many orders of magnitude smaller than masses of their family partners, leptons and quarks. . . . The most natural possibility of the explanation of the smallness of the neutrino masses gives us the seesaw mechanism of the neutrino mass generation. This beyond the Standard Model mechanism connects smallness of neutrino masses with the violation of the total lepton number at a large scale and Majorana nature of neutrino masses. If it will be established that neutrinos with definite masses are Majorana particles it will be strong argument in favor of the seesaw origin of neutrino masses.
Investigation of the neutrinoless double beta-decay of nuclei is the only practical way which could allow to proof that neutrinos are Majorana particles.
Theorists tend to prefer the assumption that the neutrino and anti-neutrino are identicial, and hence that a process known as neutrinoless double beta decay is possible.
The Experimental Constraints On Neutrinoless Double Beta Decay
One experiment by H. V. Klapdor-Kleingrothaus (Heidelberg-Moscow) published in 2001 claimed to see neutrinoless double beta decay experimentally, and claimed six sigma support for that conclusion by 2006, but the experiment has not been successfully replicated in three other completed attempts to do so, and has been subject to considerable criticism in the discipline.
More than a dozen current or proposed experiments that are already under construction or will commence construction in the next few years, are looking for signs of neutrinoless double beta decay.
The predicted frequency of neutrinoless double beta decay in a simple Majorana mass scenario is a product of the three neutrino mass eignenstates and the correspoding PMNS matrix elements. A 2010 recap of the theory is found here. The key number that is produced using these estimates is on the order of 0.2-0.6 eV according to H. V. Klapdor-Kleingrothaus which corresponds to effective Majorana masses. A larger number would yield a higher (and presumably easier to observe) decay rate, while a smaller number would yield a lower (and presumably hard to observe) decay rate.
An effective Majorana mass of this scale would imply absolute neutrino masses that are much greater than the experimentally established values for the differences in mass between the three neutrino mass eigenstates by a couple of orders of magnitude, and hence, a nearly degenerate set of neutrino mass eigenstates, a result that seems like a poor fit to a measured value of theta 12 in the PMNS matrix that is about ten times as large as theta 13 in the PMNS matrix - since big differences in transition matrix values seem to have some association with big differences in mass between the particles in question.
Experiments that are underway would increase the sensitivity of the experiments to Majorana masses more than ten times as small as that of Klapdor-Kleingrothaus, making is possible to rule out or confirm that finding. Direct neutrino detection experiments, such as Ice Cube, which just went on line in Antarctica, also provide measurements of neutrino properties that can constrain the theoretically expected values for neutrinoless double beta decay in Majorana mass neutrino models. (See also here setting out the experimental agenda for neutrino research in 2004 through about 2014).
While current experiments establish only the relative mass differences between neutrino eigenstates, rather than absolute masses, if the absolute values are on the same order of magnitude as those differences (on the order of 0.003 eV for the first and second mass eigenstate gaps and 0.05 eV for the second and third mass eigenstate gap), then they are much lower than the Klapdor-Kleingrothaus effective Majorana neutrino mass estimate, and would seem to be inconsistent with a Majorana neutrino mass in anything but a normal mass hierarchy (a first generation neutrino mass lighter than a second generation neutrino mass which is lighter in turn than a third generation neutrino mass). Astronomy data also place significant minimum values on neutrinoless double beta decay rates (the linked article also remarks on the very strict current experimental limitations on the magnetic moment of neutrinos which disfavors the possibility that they may be composed of charged preons).
Neutrinoless Double Beta Decay In SUSY Models
Neutrinoless double beta decay experiments also constrain SUSY models which need to have a characteristic SUSY scale on the order of 1 TeV to fit that Klapdor-Kleingrothaus measurement, or smaller if that measurement is not replicated. (Larger decay values have been pretty well ruled out, and by implication, characteristic SUSY scales in SUSY models with Majorana mass of more than 1 TeV, which should be within the power of the LHC to detect, are also disfavored.)
If neutrinoless double beta decay is ten times more rare than that measurement, this would imply a characteristic SUSY scale on the order of 630 GeV, which is a scale that is likely to be ruled out or confirmed at LHC around the same time that that neutrinoless double beta decay experiment results with that precision are available. Neutrinoless double beta decay rates that were thirty or forty times as small as the claimed Klapdor-Kleingrothaus measurement in a SUSY matter would bring the characteristic SUSY scale so low that it would be inconsistent with current LHC bounds.
Of course, nimble theorists can always come up with some variant theory that would escape these bounds (see, e.g. this paper from 2007 with Dirac neutrino masses in a SUSY variant). Indeed, the sheer number of beyond the Standard Model proposals to deal with neutrino properties are immense, although the many are simply slight variants on the same themes. But, the bound on SUSY theories from neutrinoless double beta decay is notable because it is experimentally independent of the particle accelerator driven bounds on the masses of the lighest supersymmetric particles, and because non-detection of neutrinoless double beta decay favors smaller SUSY scales, while non-detection of supersymmetric particles at particle accelerators favor larger characteristic SUSY scales. Taken together, neutrinoless double beta decay experiments and the LHC operate as a vice squeezing SUSY parameter space in opposite directions.
Neutrinos Lack Majorana Mass
My personal prediction is that we will eventually establish bounds on absolute neutrino eigenstate mass and bounds on Majorana mass from a failure to detect neutrinoless double beta decay that will together establish definitively that neutrinos have Dirac masses, just like all other Standard Model fermions and as a result of the same mechanism despite the fact that neutrino masses are much smaller than other Dirac masses, that neutrinos and antineutrinos are not the same thing.
This prediction is driven mostly by the pivotal role that the distinction between a neutrino and antineutrino plays in maintaining lepton number conservation (which has never been observed to be violated experimentally and produced large numbers of valid predictions about decay patterns) that motivated their predicted existence in the first place.
A fortiori, this prediction also assumes that SUSY models with Majorana neutrino masses are also wrong. There are other reasons to find the remaining range of SUSY parameter space to be implausible, but this is another one which is quite strict.
There Are No Sterile Neutrinos or Fourth Generation Fermions
A finding that neutrinos lack Majorana mass would not necessarily rule out the possibility that there are right handed neutrinos (aka sterile neutrios) with Dirac mass that give rise to left handed neutrino mass via a seesaw mechanism. The Standard Model assumed that neutrinos had no mass at all, so it is indeterminate as to how this issue is resolved, and its other predictions are largely decoupled from it.
Precision electroweak measurements suggest that fourth generation left handed neutrinos of less than 45 GeV are ruled out, which would be so far in excess of the other three neutrino masses that it makes the entire notion of a fourth generation of Standard model fermions seem implausible. But, because right handed neutrinos would not interact with the weak force, precision electroweak measurements can't rule them out or say much of anything about what masses they might have, although seesaw models tend to favor right handed neutrinos that are much heavier than left handed neutrinos.
Theories with heavy sterile neutrinos draw succor from the perceived need for a fairly heavy dark matter candidate, although direct dark matter searches and astronomy data are incresingly narrowing the experimental window in which such heavy dark matter particles could exist. They also find support from the fact that there are four permutations at each generation of every charged fermion in the Standard Model (LH particle, LH antiparticle, RH particle, RH antiparticle), so the existence of only a LH particle and RH antiparticle seems to leave the neutrino column of the chart of Standard Model particles with gaps, and it is hard to rule out the presence of something in those gaps because a right handed neutrino would be so inherently weakly interacting apart from its gravitational interactions, just as hypothesized dark matter.
But, heavy right handed neutrinos would also contradict the pattern for all of the charged fermions of the Standard Model in which the right handed and left handed versions of the particle and the right handed and left handed version of the antiparticle all had the same mass.
Very heavy right handed neutrinos also seem out of line with the example of the Z boson, which is its own antiparticle, which has a mass only marginally greater than that of the W+ boson which has the W- boson as an antiparticle, with all three being intimately intertwined, and the Higgs boson having a mass on the order of the sum of the three weak force boson masses. Similarly, neutrons are not dramatically heavier than protons, and electromagnetically neutral hadrons generally are not so much different in mass from electrically charged hadrons. If charge or its lack has in impact on mass, it does not seem to be a dramatic influence.
Also, since baryogenesis and leptogenesis scenarios generally assume that quarks and leptons have their origins in weak force decays, any hypothesis with right handed neutrinos must also come up with a leptogenesis scenario specific to them.
My personal prediction, although I make it with far less confidence than I do when predicting that neutrinos lack Majorana mass, is that there are no right handed neutrinos.
The PMNS Matrix has a CP violating phase
There are good reasons from quark-lepton complementarity to suspect that that PMNS matrix has a CP violating phase complementary to the CP violating phase in the CKM matrix.
There also seems to be preliminary evidence for the existence of such a phase at the MINOS experiment where the profiles of neutrinos and antineutrinos seem to be different. (Incidentally, CP violation would also seem to disfavor Majorana neutrino theories, since if the particle and antiparticle are identical, they shouldn't exhibit different behavior.)
I expect that CP violations will be confirmed in the PMNS matrix, in W boson mediated interactions, but not Z boson mediated interactions, with a phase complementary in some way to that of the CKM matrix CP violating phase.
Conclusion Regarding Predictions
My predictions are generically "dull" from a theorist's perspective. They leave the mass generation mechanism for neutrinos in a "black box", they predict no new particles to serve as dark matter candidates (neutrino condensates or perhaps stable glueballs begin to look attractive as dark matter candidates in this scenario), and they predict no new kinds of particle interactions.
They also throw the vast majority of the theoretical output on neutrino physics and models that call for right handed neutrinos, Majorana neutrinos, or seesaw mechanisms into the dustbin. Basically, tens of thousands of fundamental physics papers over the last decade are counterfactual flights of fancy in this scenario.
This approach would seem generically to leave conventional grand unified theories and theories of everything overconstrained. Most predict something more than the Standard Model or are inconsistent with experiment. A nice summary of the data points these models try to fit can be found here. (Footnote, I hadn't noticed before that the not quite running coupling constant scale of the Standard Model is a couple of orders of magnitude lower than the SUSY GUT scale, which would make concerns about very high energy scale breakdowns of the Standard Model with a Higgs boson of the experimentally suggested mass less intense).