Neutrinos do have mass, as it has been demonstrated by several oscillation experiments. The question is what kind of mass they have. All the other fermions in the Standard Model have a Dirac-type mass, but the neutrino, being chargeless, can have a Majorana-type mass, namely, be indistinguishable from its own antiparticle. Majorana neutrinos provide an attractive explanation for the smallness of neutrino masses, the so-called seesaw mechanism. Besides, a Majorana nature of neutrinos would imply a violation of the lepton number, since a lepton can be converted into an antilepton. Lepton number violation is one of the basic ingredients, together with CP violation, to explain the asymmetry between matter and antimatter in our universe.
If the neutrino is the same as its own antiparticle, the neutrinoless double beta nuclear decay is allowed. In a double beta decay, two neutrons in the same nucleus convert into protons, emitting an electron and an antineutrino each. If the neutrino is the same particle as the antineutrino, the two antineutrinos can annihilate and just two electrons are emitted. The reverse has also been demonstrated, namely that the existence of such a decay implies the Majorana nature of neutrinos.
The signal of a neutrinoless double beta decay is a peak in the kinetic energy of the outcoming electrons. If neutrinos are present, part of the available energy is carried by them, so the spectrum is a continuous distribution with endpoint at Qββ. On the contrary, in the neutrinoless case the electrons have all the available energy of the decay.
S. Elliott, Recent Progress in Double Beta Decay, 2012.
J. J. Gómez-Cadenas et al., The search for neutrinoless double beta decay, Riv. Nuovo Cim. 35 (2012) 29-98.
J. J. Gómez-Cadenas et al., Sense and sensitivity of double beta decay experiments, JCAP 1106 (2011) 007.
S. M. Bilenky, Neutrinoless double beta decay, Phys. Part. Nucl. 41:690-715, 2010.
F. T. Avignone, S. R. Elliott and J. Engel, Double Beta Decay, Majorana Neutrinos, and Neutrino Mass, Rev. Mod. Phys. 80:481-516, 2008.
S. R. Elliott and P. Vogel, Double Beta Decay, Ann. Rev. Nucl. Part. Sci. 52:115-151, 2002.