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Description of the $2ννββ$ decay within a fully renormalized pnQRPA approach with a restored gauge symmetry

A many body Hamiltonian involving the mean field for a projected spherical single particle basis, the pairing interactions for alike nucleons and the dipole-dipole proton-neutron interactions in the particle-hole ($ph$) and particle-particle ($pp$) channels is treated by the projected gauge fully renormalized proton-neutron quasiparticle random phase approximation (PGFRpnQRPA) approach. The resulting wave functions and energies for the mother and the daughter nuclei are used to calculate the $2νββ$ decay rate and the process half life. For illustration, the formalism is applied for the decay $^{100}$Mo$ \to$ $^{100}$Ru. The results are in good agreement with the corresponding experimental data. The Ikeda sum rule ($ISR$) is obeyed. The gauge projection makes the $pp$ interaction inefficient.