The observations of neutrino oscillations have shown that the neutrinos have mass and have determined their mass splittings. The existence of zero-neutrino double beta decay will show that the neutrino is its own anti-particle, and the half-life will determine the absolute mass scale. The rate for this decay is proportional to the square of nuclear matrix elements that must be calculated in order to extract the mass. I will discuss the model dependence of the matrix elements for two-neutrino decay, and zero-neutrino decay involving light and heavy neutrinos. These involve a variety of two-body operators who's expectation values depend on the particle-hole and particle-particle (pairing) correlations in the nuclear wave functions. These correlations will be discussed in terms of a shell model configuration space (a horizontal truncation) together with corrections involving other orbitals (a vertical expansion). The results can related to other nuclear properties including isospin symmetry, Gamow-Teller beta decay, and the odd-even oscillations in the binding energies.

Argonne Physics Division Colloquium Schedule