Renormalized QRPA and double beta decay: a critical analysis

The proton-neutron monopole Lipkin model, which exhibites the properties which are relevant for the description of the double beta decay ($\beta \beta$) transitions, is solved exactly. The exact results are compared with the ones obtained by using the Quasiparticle Random Phase (QRPA) and renormalized QRPA (RQRPA) approaches. It is shown that the RQRPA violates the Ikeda Sum Rule and that this violation may be common to any extension of the QRPA which neglects scattering terms in the participant one-body operators as well as in the Hamiltonian. This finding remains valid even when exact wave function are used to compute two-quasiparticle leading order terms of the transition operators. It underlines the need of additional developments before the RQRPA could be adopted as a reliable tool to compute $\beta \beta$ processes.