The Standard Model of Particle Physics without Majorana neutrinos, does not permit a weak force mediated beta decay channel called "neutrinoless double beta decay." In the Standard Model, a quantity called "lepton number" is conserved and that conservation of lepton number prohibits this decay from taking place (see, e.g. here).
The Importance of Neutrinoless Double Beta Decay To BSM Theories of Physics
Many beyond the standard model theories, including a popular variation of the Standard Model and extensions of the Standard Model with Majorana neutrinos (i.e. neutrinos which are their own antiparticles), do permit neutrinoless double beta decay and even make fairly specific predictions about how common this kind of decay must be relative to other kinds of weak force decays, like ordinary beta decay. Many beyond the Standard Model theories also argue that the quantity baryon minus lepton number (B-L), rather than baryon number (B) and lepton number (L) independently, is what is conserved in the most extreme conditions.
Theorists are motivated to do this by cosmology. We know that there is more matter than antimatter in the universe, presumably resulting from an asymmetry of matter and anti-matter production. But, "it is impossible to generate the observed asymmetry if the standard model of particle interactions is strictly adhered to," as noted in this pre-print. Yet, terrestrial scale experiments have failed to generate any evidence of B or L number conservation violations. Conserving B-L, while not conserving either B or L independently in extreme conditions, provides one way of generating the observed asymmetry in a fairly theoretically conservative way.
An broad review of the theoretical and experimental considerations related to neutrinoless double beta decay, can be found in a December 19, 2011 post at this blog. A fairly accessible recent review in the scientific literature from earlier this year, which also illustrates how improved precision in determining other physical constants related to neutrinos can influence the precision of neutrinoless double beta decay searches, can be found here. The many "medium sized physics" experiments currently in progress to study this are summarized here.
Neutrinoless double beta decay research is probably the single most important kind of fundamental physics research going on right now outside the Large Hadron Collider (LHC) in terms of excluding or supporting various beyond the standard model physics theories. A factor of ten lower limit on the maximum rate at which it can occur would rule out a large swath of currently popular beyond the Standard Model theories. (Many of the other cutting edge non-LHC fundamental physics experimental research agendas also involve studies of neutrino properties, as this is one of the least well confirmed and understood parts of the Standard Model.)
For example, sufficiently rare neutrinoless double beta decays strongly constrain much of the SUSY parameter space in a manner independent of the constraints of LHC SUSY searches. The more rare neutrinoless double beta decays are, the lower the characteristic SUSY energy scale must be. The Heidelberg-Moscow result in most SUSY models implies a characteristic SUSY energy scale of about 1 TeV, and a lower neutrinoless double beta decay rate implies a lower characteristic SUSY energy scale. Meanwhile the minimum SUSY energy scale in most SUSY models is bounded by the non-detection of lighter SUSY particles at the Large Hadron Collider. Non-detection of SUSY predicted phenomena in the two kinds of experiments squeeze the characteristic SUSY energy scale in opposite directions.
Similarly, a theoretical analysis pre-print published a couple of weeks before the latest more stringent bounds on neutrinoless double beta decay rates reported in this post found that experiment constrained theoretical sterile neutrinos to be less than 10 GeV, considerable lighter than many proposed sterile neutrino WIMP (i.e. dark matter) possibilities; electro-weak experimental bounds have ruled out weakly interacting neutrinos other than the three known generations of them to not less than about 45 GeV.
The combined results already come close to ruling out any SUSY model in which neutrinos have Majorana rather than Dirac mass, i.e. most SUSY models. And, most String theorists see SUSY as a low energy approximation of any String Theory, so an exclusion of large numbers of possible SUSY models also dramatically narrows the parameter space of String Theory.
The Weak And Contested Evidence For Neutrinoless Double Beta Decay
No widely accepted study has concluded that it has discovered neutrinoless double beta decay, although there is one non-replicated claim that it has been discovered.
More than ten years ago, the collaboration behind the Heidelberg-Moscow Double Beta Decay Experiment controversially claimed to have discovered neutrinoless double beta decay using germanium-76 isotopes. But now, the EXO-200 researchers say, their new data makes it highly unlikely that those earlier results were valid.The Heidlberg-Moscow experiment claimed finding would imply an effective Majorana mass of 0.2-0.6 eV (reference here). Other experiments in the intervening decade, including three failed replication attempts, have likewise cast doubt on the Heidelberg-Moscow result which has not received wide acceptance.
Astronomy data as well as data from experiments like the Heidleberg-Moscow experiment and EXO-200 also place stringent limits on the maximum rate of neutrinoless double beta decay, as well as disfavoring any structure insider neutrinos derived from hypothetical component charged particles that would be one kind of preon (a general term that applies to any component of a particle that is fundamental in the Standard Model).
The New Experimental Evidence From Two New Experiments Using Liquid Xenon 136
A new study by EXO-200, which looks at decays from atomically unstable liquid xenon-136, has put even more stringent bounds on the maximum rate at which neutrinoless double beta decay can occur without finding any convincing examples of it. The published version of the new study is: M. Auger, et al., "Search for Neutrinoless Double-Beta Decay in 136Xe with EXO-200." Physical Review Letters, 2012 [link]. Preprint here.
[S]even months of finding nothing means that the half-life [of neutrinoless double decay producing events] cannot be shorter than 1.6 × 1025 years, or a quadrillion times older than the age of the universe. . . . The new data [also] suggest that a neutrino cannot be more massive than about 0.140 to 0.380 electron volts (eV, a unit of mass commonly used in particle physics); an electron, by contrast, is about 500,000 eV, or about 9 × 10-31 kilograms.Thus, EXO-200 has ruled out about half of the Heidleberg-Moscow parameter space in the first seven months and is expected to continue its search for many years to come. According to the EXO-200 paper: "The present result contradicts [Heidleberg-Moscow (2006)] at 68% CL (90% CL) for all (most) matrix element calculations considered[.]"
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 Heidelberg-Moscow experiment prediction, sometimes also described as the Klapdor-Kleingrothaus result, after the lead experimenter in that experiment.
An almost identical experiment called KamLAND-Zen also reported its results last week (also here confirming the double neutrino beta decay results form EXO-200), and similarly found no evidence of neutrinoless double beta decay in its first five months, although its limits were not quite as strict as those reported by EXO-200: "a lower limit on the ordinary (spectral index n = 1) Majoron-emitting decay half-life of Xe-136 is obtained as T_{1/2}^{0\nu\chi^{0}} > 2.6 x 10^{24} yr at 90% C.L., a factor of five more stringent than previous limits." (The EXO-200 result of >1.6*10^25, by implication, is 32 times more stringent that previous limits.) The data, were they pooled, would surely support a limit lower than the 1.6*10^25 year limit of the EXO-200 experiment by itself, and also a lower bound on potential Majorana mass in neutrinos.
In a few years, the two experiments should provide enough data to conclusively contradict the Heidleberg-Moscow experiment and simultaneously contrain the neutrinoless double beta decay rate to a level sufficiently low to rule out or confirm a wide variety of beyond the Standard Model theories. A confirmation would be the first serious experimental challenge to the Standard Model since neutrino mass was discovered to exist.
My Own Expectations
I am on record as having predicted that neutrinoless double beta decay does not happen, that neutrinos do not have Majorana mass, and that neutrinos are not Majorana particles, for a mix of experimental and theoretical reasons.
For what it is worth, I also don't think that there are B number or L number conservation violations in the current universe - my own pet conjecture regarding matter-antimatter asymmetry is that at the time of the Big Bang, an equal amount of matter and antimatter was generated, but the matter mostly went "forward" in time from the Big Bang, while the antimatter mostly went "backward" in time from the Big Bang, where there is an antimatter dominated universe where time is perceived to flow in the opposite direction from the one that we perceive it to flow in our own universe. But, this is undetectable, since the Big Bang is between the two universes. I don't claim that this conjecture is widely shared in the field, however.
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