Nuclear Matrix Elements for the neutrinoless Double Beta Decay: a critical Review

If the neutrino is a Majorana particle, the neutrinoless double beta decay can give an absolute scale for the neutrino mass. In the Standard Model, the neutrinoless double beta decay is not allowed but it is allowed in most Grand Unified Theories (GUT's). The neutrino must be identical with its antiparticle and must have a mass to allow the neutrinoless double beta decay. Apart of one claim that the neutrinoless double beta decay in (76)Ge is measured, one has only tipper limits for this transition probability. But even the upper limits allow to give upper limits for the electron Majorana neutrino mass and upper limits for parameters of GUT's 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 GUT's. For that one has to assume that the specific mechanicsm is the leading one for the neutrinoless double beta decay and one has to be able to calculate reliably the corresponding nuclear matrix elements. In the present contribution, one discusses the accuracy of the present status of calculating the nuclear matrix elements.