Two-neutrino double-beta decay Fermi transition and two-nucleon interaction

An exactly solvable model for a description of the two-neutrino double beta decay transition of the Fermi type is considered. By using perturbation theory an explicit dependence of the two-neutrino double beta decay matrix element on the like-nucleon pairing, particle-particle and particle-hole proton-neutron interactions by assuming a weak violation of isospin symmetry of Hamiltonian expressed with generators of the SO(5) group. It is found that there is a dominance of double beta decay transition through a single state of the intermediate nucleus. Then, an energy weighted sum rule connecting Delta Z=2 nuclei is presented and discussed. It is suggested that this sum rule can be exploited to study the residual interactions of the nuclear Hamiltonian.