The nuclear matrix elements that govern the rate of neutrinoless double beta decay must be accurately calculated if experiments are to reach their full potential. Theorists have been working on the problem for a long time but have recently stepped up their efforts as ton-scale experiments have begun to look feasible. Here we review past and recent work on the matrix elements in a wide variety of nuclear models and discuss work that will be done in the near future. Ab initio nuclear-structure theory, which is developing rapidly, holds out hope of more accurate matrix elements with quantifiable error bars.

We review the impact of nuclear forces on matter at neutron-rich extremes. Recent results have shown that neutron-rich nuclei become increasingly sensitive to three-nucleon forces, which are at the forefront of theoretical developments based on effective field theories of quantum chromodynamics. These developments include the formation of shell structure, the spectroscopy of exotic nuclei, and the location of the neutron drip line. Nuclear forces also constrain the properties of neutron-rich matter, including the neutron skin, the symmetry energy, and the structure of neutron stars. First, we review our understanding of three-nucleon forces and show how chiral effective field theory makes unique predictions for many-body forces. Then, we survey results with three-nucleon forces in neutron-rich oxygen and calcium isotopes and neutron-rich matter, which have been explored with a range of many-body methods. Three-nucleon forces therefore provide an exciting link between theoretical, experimental, and observational nuclear physics frontiers.

We study in this Letter the neutrinoless double beta decay nuclear matrix elements (NME’s) in the framework of the interacting shell model. We analyze them in terms of the total angular momentum of the decaying neutron pair and as a function of the seniority truncations in the nuclear wave functions. This point of view turns out to be very adequate to gauge the accuracy of the NME’s predicted by different nuclear models. In addition, it gives back the protagonist role in this process to the pairing interaction, the one which is responsible for the very existence of double beta decay emitters. We show that low seniority approximations, comparable to those implicit in the quasiparticle RPA in a spherical basis, tend to overestimate the NME’s in several decays.

We perform state-of-the-art large-scale shell-model calculations of the structure factors for elastic spin-dependent WIMP scattering off ${}^{129,131}\mathrm{Xe}$, ${}^{127}\mathrm{I}$, ${}^{73}\mathrm{Ge}$, ${}^{19}\mathrm{F}$, ${}^{23}\mathrm{Na}$, ${}^{27}\mathrm{Al}$, and ${}^{29}\mathrm{Si}$. This comprehensive survey covers the nonzero-spin nuclei relevant to direct dark matter detection. We include a pedagogical presentation of the formalism necessary to describe elastic and inelastic WIMP-nucleus scattering. The valence spaces and nuclear interactions employed have been previously used in nuclear structure calculations for these mass regions and yield a good spectroscopic description of these isotopes. We use spin-dependent WIMP-nucleus currents based on chiral effective field theory at the one-body level and including the leading long-range two-body currents due to pion exchange, which are predicted in chiral effective field theory. Results for all structure factors are provided with theoretical error bands due to the nuclear uncertainties of WIMP currents in nuclei.