Towards a complete description of the neutrinoless double beta decay

Recent trends in the analyses of neutrino oscillations and cosmological data suggest that the neutrino mass hierarchy is likely to be normal. This development has consequences for the neutrinoless double beta decay, namely it is more likely to have competition between the standard mass mechanism and other beyond Standard Model mechanisms that may appear above TeV scale. Here we discuss some venues of distinguishing between these potentially competing mechanisms based on shell model nuclear matrix elements.