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.

A self-consistent calculation is performed within an equation of motion method to study the electric dipole response in the neutron rich ${}^{132}\mathrm{Sn}$ with special attention at the low-energy excitations associated with the pygmy dipole resonance. A basis of one- plus two-phonon states is constructed by solving iteratively a set of equations of motion and adopted to bring the nuclear Hamiltonian to diagonal form. The phonons are generated in a Tamm-Dancoff approximation using a Hartree-Fock basis. The Hamiltonian is composed of an intrinsic kinetic term and an optimized two-body chiral potential. If complemented with a weak phenomenological density dependent term, such a potential improves substantially the description of the dipole response as compared to other realistic interactions. The two-phonon states play an important role in determining the fine structure of the giant dipole resonance and in enhancing the fragmentation of the strength at low energy, in fair agreement with the available data. Insights into the nature of the excitations building up the pygmy resonance are gained from the analysis of the phonon composition and shell structure of the states.

A Bogoliubov quasiparticle formulation of an equation-of-motion phonon method, suited for open-shell nuclei, is derived. Like its particle-hole version, it consists of deriving a set of equations of motions whose iterative solution generates an orthonormal basis of $n$-phonon states ($n=0,1,2,...$), built of quasiparticle Tamm-Dancoff phonons, which simplifies the solution of the eigenvalue problem. The method is applied to the open-shell neutron-rich ${}^{20}\text{O}$ for illustrative purposes. A Hartree-Fock-Bogoliubov canonical basis, derived from an intrinsic two-body optimized chiral Hamiltonian, is used to derive and solve the eigenvalue equations in a space encompassing a truncated two-phonon basis. The spurious admixtures induced by the violation of the particle number and the center-of-mass motion are eliminated to a large extent by a Gram-Schmidt orthogonalization procedure. The calculation takes into account the Pauli principle, is self-consistent, and is parameter free except for the energy cutoff used to truncate the two-phonon basis, which induces an increasing depression of the ground state through its strong coupling to the quasiparticle vacuum. Such a cutoff is fixed so as to reproduce the first $1^{-}$ level. The two-phonon states are shown to enhance the level density of the low-energy spectrum, consistently with the data, and to induce a fragmentation of the $E1$ strength which, while accounting for the very low $E1$ transitions, is not sufficient to reproduce the experimental cross section in the intermediate energy region. This and other discrepancies suggest the need of including the three-phonon states. These are also expected to offset the action of the two phonons on the quasiparticle vacuum and, therefore, free the calculation from any parameter.

Background: The electric dipole strength detected around the particle threshold and commonly associated with the pygmy dipole resonance offers unique information on neutron skin and symmetry energy, and is of astrophysical interest. The nature of such a resonance is still under debate.