Shell model studies of the $^{130}Te$ neutrinoless double-beta decay

Most uncertainties regarding the theoretical study of the neutrinoless double-beta decay are related to the accuracy of the nuclear matrix elements that appear in the expressions of the lifetimes. We calculate the nuclear matrix elements for the $0\nu\beta\beta$ decay of $^{130}Te$ in a shell model approach, using a recently proposed effective Hamiltonian. To ensure the reliability of the results, we investigate this Hamiltonian by performing calculations of spectroscopic quantities and comparing them to the latest experimental data available, and we analyze the $2\nu\beta\beta$ and the $0\nu\beta\beta$ decay nuclear matrix elements of $^{136}$Xe. Finally, we report new nuclear matrix for the $^{130}$Te considering the light neutrino exchange and heavy neutrino exchange mechanisms, alongside with an overview of some recent values reported in the literature.