Neutrinoless $\beta\beta$-Decay in DCEQTDA

We have recently developed a nuclear model, which is a natural extension of the $pn$-QRPA model, specially designed to describe double charge exchange (DCE) processes generated by two-body DCE transition operators. It is based on the Quasiparticle Tamm-Dancoff Approximation (QTDA) for $pn$ and $2p2n$ excitations in intermediate and final nuclei, respectively, and will be called DCEQTDA. As such, this model, having the same number of free parameters as the $pn$-QRPA, also brings into play the excitations of four quasiparticles to build up the final nuclear states, which are then used to evaluate the nuclear matrix elements (NMEs) for all $0^+$ and $2^+$ final states, including resonances, and not just for the ground state as in $pn$-QRPA. In addition, it allows us to evaluate: (a) the values of $Q_{\beta\beta}$, (b) the excitation energies in final nuclei, and (c) the DCE sum rules, which are fulfilled in the DCEQTDA. So far, this model has been used mainly to calculate double beta decays with the emission of two neutrinos ($2\nu\beta\beta$-decay). Here, we extend it to the study of these processes when no neutrinos are emitted ($0\nu\beta\beta$-decay), evaluating them in a series of nuclei, but paying special attention to (i) $^{76}$Se, which have been measured recently in the GERDA and MAJORANA experiments, and (ii) $^{124}$Te, for which the first direct observation of the double electron capture $2\nu$ has been performed with the XENON1T dark matter detector. We obtain good agreement with the data for both the ground state and the excited states. The validity of the DCEQTDA model is checked by comparing the calculation with the experimental data for the $2\nu\beta\beta$ NMEs, and for the $Q_{\beta\beta}$, in a series of nuclei.