Nuclear structure and the double beta decay

In the Standard Model, the neutrinoless double beta decay is not allowed but it is allowed in most Grand Unified Theories (GUTs). The neutrino must be a Majorana particle (identical with its antiparticles) and must have a mass to allow the neutrinoless double beta decay. Apart from one claim that the neutrinoless double beta decay in 76Ge is measured, one has only upper limits for this transition probability. But even the upper limits allow us to give upper limits for the electron Majorana neutrino mass and upper limits for parameters of GUTs and the minimal R-parity violating supersymmetric model. One further can give lower limits for the vector boson mediating mainly the right-handed weak interaction and the heavy mainly right-handed Majorana neutrino in left–right symmetric GUTs. For that one has to assume that the specific mechanism is the leading one for the neutrinoless double decay and one has to be able to calculate reliably the corresponding nuclear matrix elements. In the present contribution, we discuss the accuracy of the present status of calculating the nuclear matrix elements and the corresponding limits of GUTs and supersymmetric parameters.