In this work, we approach for the first time the task of performing a shell-model calculation of the matrix element for neutrinoless double-$\beta$ decay, within a fully-consistent framework where the expressions of the nuclear Hamiltonian and of the decay operators have been derived through chiral perturbation theory. More precisely, the effective shell-model Hamiltonian and all transition operators have been constructed by way of many-body perturbation theory, and then employed to calculate both spectroscopic properties of the nuclei involved in the decays under our consideration—namely ${}^{48}\mathrm{Ca}$, ${}^{76}\mathrm{Ge}$, and ${}^{82}\mathrm{Se}$—as well as the nuclear matrix elements of the electromagnetic and neutrinoless double-$\beta$ decays. We also present a study of the convergence properties of the calculated matrix elements in order to provide the elements for an estimate of the theoretical uncertainty.

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• Beta decay
• Electroweak interactions in nuclear physics
• Nuclear forces
• Nuclear structure & decays
• Nuclear many-body theory
• Shell model