KamLAND-Zen is a group of buddhist monks studying a balloon filled with the xenon isotope Xe136. That isotope has a very long lifetime, of order 10^21 years, and undergoes the lepton-number-conserving double beta decay Xe136 → Ba136 2e- 2νbar. What the monks hope to observe is the lepton violating neutrinoless double beta decay Xe136 → Ba136+2e, which would show as a peak in the invariant mass distribution of the electron pairs near 2.5 MeV. No such signal has been observed, which sets the limit on the half-life for this decay at T>1.1*10^26 years.
The neutrinoless decay is predicted to occur if neutrino masses are of Majorana type, and the rate can be characterized by the effective mass Majorana mββ (y-axis in the plot). That parameter is a function of the masses and mixing angles of the neutrinos. In particular it depends on the mass of the lightest neutrino (x-axis in the plot) which is currently unknown. Neutrino oscillations experiments have precisely measured the mass^2 differences between neutrinos, which are roughly (0.05 eV)^2 and (0.01 eV)^2. But oscillations are not sensitive to the absolute mass scale; in particular, the lightest neutrino may well be massless for all we know. If the heaviest neutrino has a small electron flavor component, then we expect that the mββ parameter is below 0.01 eV. This so-called normal hierarchy case is shown as the red region in the plot, and is clearly out of experimental reach at the moment. On the other hand, in the inverted hierarchy scenario (green region in the plot), it is the two heaviest neutrinos that have a significant electron component. In this case, the effective Majorana mass mββ is around 0.05 eV. Finally, there is also the degenerate scenario (funnel region in the plot) where all 3 neutrinos have very similar masses with small splittings, however this scenario is now strongly disfavored by cosmological limits on the sum of the neutrino masses (e.g. the Planck limit Σmν < 0.16 eV).
As can be seen in the plot, the results from KamLAND-Zen, when translated into limits on the effective Majorana mass, almost touch the inverted hierarchy region. The strength of this limit depends on some poorly known nuclear matrix elements (hence the width of the blue band). But even in the least favorable scenario future, more sensitive experiments should be able to probe that region. Thus, there is a hope that within the next few years we may prove the Majorana nature of neutrinos, or at least disfavor the inverted hierarchy scenario.