Solution of the Dyson equation for nucleons in the superfluid phase

Abstract We investigate the role the interweaving of surface vibrations and nucleon motion has on Cooper pair formation in spherical superfluid nuclei. A quantitative calculation of the state-dependent pairing gap requires to go beyond the quasiparticle approximation, treating in detail the breaking of the single-particle strength and of the associated poles. This is done solving self-consistently the Dyson equation, including both a bare nucleon–nucleon interaction (which for simplicity we choose as a monopole-pairing force of constant matrix elements g ) and an induced interaction arising from the exchange of vibrations (calculated microscopically in QRPA) between pairs of nucleons moving in time reversal states. Both the normal and anomalous density Green functions are included, thus treating self-energy and pairing processes on equal footing. We apply the formalism to the superfluid nucleus 120 Sn. Adjusting the value of g so as to reproduce, for levels close to the Fermi level, the empirical odd–even mass difference ( Δ ≈1.4 MeV), it is found that the pairing gap receives about equal contributions from the monopole-pairing force and from the induced interaction. This result is also reflected in the fact that if one were to reproduce the observed Δ allowing the nucleons to interact only through the bare monopole-pairing force, a value of g ≈0.233 MeV (≈28/ A MeV) is needed, 50% larger than the value g ≈0.166 MeV (≈20/ A MeV) needed in the full calculation. To keep in mind that the bare and the induced pairing contributions to Δ enter the corresponding equations in a very nonlinear fashion. It is furthermore found that self-energy processes reduce the contribution of the phonon induced interaction to the pairing gap by ≈20% as compared to the value obtained by only phonon exchange without taking into account the breaking of the single-particle strength.