The Standard Model (SM) of electroweak and strong interactions cannot be considered a complete and ultimate theory. Large number of free parameters (>15), three independent coupling constants and three independent gauge groups indicate a more general theory broken down to what we see by symmetry violations at low energy. Neutrinos are described as massless fermions, whereas there is stronger and stronger experimental evidence contradicting this assumption. At present there are five independent indications of finite neutrino masses:
To make progress in this field one has to look for observables that could give indication of physics beyond the SM. Here the nucleus is a very useful laboratory where the fundamental conservation laws can be tested. The advantage of the nucleus is to have a wide spectrum of different initial and final states for transitions and reactions. These states can be chosen in such a way that the transitions are forbidden without violation of symmetries or that such a violation produces an appreciable different observables like the angular distribution of the electrons in the beta decay.
Specifically the double beta decay and R-parity violating supersymmetric (SUSY) processes are subjects of wide interest. From the nonobservability of the neutrinoless mode of the double beta decay one can deduce limits of the non-standard model parameters, like the structure of the neutrino mass matrix or scale of left-right symmetry breaking what could confirm or rule out some Grand Unified Theory (GUT) scenarios. The currently running experiments, like the Heidelberg-Moscow collaboration can push the neutrino mass searches into the sub-eV range. But while an experimental limits on the half-life of the neutrinoless mode of the double beta decay is free from theoretical assumptions, the limit on the neutrino mass is not. It depends strongly on nuclear structure theory. Therefore one of the more important aims of future researches is connected with development of reliable nuclear models for such phenomena. Then one will be able to calculate the necessary nuclear matrix elements and to test the theory on other well measured processes, like two-neutrino double beta decay or double charge exchange (DCX) with low-energy pions. The last reaction is essentially a two-nucleon sequential process, proceeding via nonanalog routes showing thus strong dependence on nucleon-nucleon correlations. Recently the idea of a d' dibaryon, a new exotic particle has been proposed with connection to the reaction mechanism.
The disadvantage of testing fundamental symmetries in nuclei is, that one has to solve the nuclear many-body problem reliably. So, in addition to calculating the observables, for example connected with the neutrinoless double beta decay, one has to calculate also other observables with the same wave functions. These observables have to be well measured and reproduced by the wave functions which one later needs for calculating the transition probabilities for neutrinoless double beta decay. Thus one has to develop methods for solving the nuclear problem and to test them for experimentally known observables to be sure that the quantities calculated for testing the SM are reliable.