For the first time, the calculation of the nuclear matrix element of the double-$\beta$ decay of ${}^{100}\mathrm{Mo}$, with and without the emission of two neutrinos, is performed in the framework of the nuclear shell model. This task is accomplished starting from a realistic nucleon-nucleon potential, then the effective shell-model Hamiltonian and decay operators are derived within the many-body perturbation theory. The exotic features which characterize the structure of Mo isotopes—such as shape coexistence and triaxiality softness—push the shell-model computational problem beyond its present limits, making it necessary to truncate the model space. This has been done with the goal to preserve as much as possible the role of the rejected degrees of freedom in an effective approach that has been introduced and tested in previous studies. This procedure is grounded on the analysis of the effective single-particle energies of a large-scale shell-model Hamiltonian, that leads to a truncation of the number of the orbitals belonging to the model space. Then, the original Hamiltonian generates a new one by way of a unitary transformation onto the reduced model space, to retain effectively the role of the excluded single-particle orbitals. The predictivity of our calculation of the nuclear matrix element for the neutrinoless double-$\beta$ decay of ${}^{100}\mathrm{Mo}$ is supported by comparison with experiments of the calculated spectra, electromagnetic transition strengths, Gamow-Teller transition strengths, and the two-neutrino double-$\beta$ nuclear matrix elements.

We perform shell-model calculations for nuclei below 100Sn, focusing attention on the two N=Z nuclei, 96Cd and 92Pd, the latter having recently been the subject of great experimental and theoretical interest. We consider nuclei for which the 0g9/2 orbit plays a dominant role and employ a realistic low-momentum two-body effective interaction derived from the CD-Bonn nucleon-nucleon potential. This implies that no phenomenological input enters our effective Hamiltonian. The calculated results for 92Pd are in very good agreement with the available experimental data, which gives us confidence in our predictions for 96Cd. An analysis of the wave functions of both 96Cd and 92Pd is performed to investigate the role of the isoscalar spin-aligned coupling.

Background: Single-charge-exchange reactions are appropriate tools to study the nuclear response to one-particle/one-hole isospin probes, gaining additional interest from the connection to beta decay. This analogy has been recently extended to second order, connecting double-charge-exchange reactions and double beta decays. Nowadays, the availability of powerful experimental setups and advanced nuclear theories and models allows one to access precious information on key nuclear structure aspects embedded in the widely sought neutrinoless double beta decay.

We report in this paper a study in terms of the nuclear shell model about the location of the calcium isotopes drip line. The starting point is considering the realistic two-body potential derived by Entem and Machleidt within chiral perturbation theory at next-to-next-to-next-to-leading order (${\mathrm{N}}^3\mathrm{LO}$), as well as a chiral three-body force at next-to-next-to-leading order (${\mathrm{N}}^2\mathrm{LO}$) whose structure and low-energy constants are consistent with the two-body potential. Then we construct the effective single-particle energies and residual interaction needed to diagonalize the shell-model Hamiltonian. The calculated two-neutron separation energies agree nicely with experiment until ${}^{56}\mathrm{Ca}$, which is the heaviest isotope whose mass has been measured, and do not show any sign of two-neutron emission until ${}^{70}\mathrm{Ca}$. We discuss the role of the choice of the model space in determining the neutron drip line, and also the dependence of the results on the parameters of the shell-model Hamiltonian.

For the first time, we approach in this work the problem of the renormalization of the Gamow-Teller decay operator for nuclear shell-model calculations by way of many-body perturbation theory, starting from a nuclear Hamiltonian and electroweak currents derived consistently by way of the chiral perturbation theory. These are the inputs we need to construct microscopically the effective shell-model Hamiltonians and decay operators. The goal is to assess the role of both electroweak currents and many-body correlations as the origins of the well-known problem of the quenching of the axial coupling constant $g_A$. To this end, the calculation of observables related to the Gamow-Teller transitions has been performed for several nuclear systems outside the ${}^{40}\mathrm{Ca}$ and ${}^{56}\mathrm{Ni}$ closed cores and compared with the available data.