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Neutrinoless double beta decay is theorized beta decay that would emit two beta particles and no neutrinos. Its detection is the goal of a number of particle physics experiments because its existence would reveal basic information about neutrinos, which are very challenging to study. It would reveal neutrinos to be Majorana particles, which implies they have mass and that they are their own antiparticle, and would be consistent with their observed oscillation. Neutrino mass is now presumed, but implies some unknown physics, and establishing them to be Majorana particles would be an additional data point. As neutrinos are an observable astrophysical phenomenon and are significant in cosmology, additional information regarding their nature has relevance to astrophysics.
Neutrinos carry off some of the energy released by beta decay and contribute to the conservation of the quantum-number totals. Some isotopes are limited to simultaneous beta decays (double beta decays) because a single beta decay would require additional binding energy whereas two decays together release some, thus provide the energy to make such decays possible. Double beta decays have been observed, but are very rare because of the required coincidence. While such a double beta decay ordinarily releases a pair of neutrinos, the neutrinos could (sometimes) annihilate during the process if they are their own antiparticle, i.e., two beta particles emitted but no neutrinos. While this would be consistent with neutrinos having mass, it would also constitute new physics because it breaks a symmetry which has been so-far unobserved: a change in one of the quantum-number totals.
Experiments to search for neutrinoless double beta decay basically carefully measure the energy of electrons emitted by selected isotope samples that are prone to double beta decay. Examples of such experiments: