Pairing matrix elements and pairing gaps with bare, effective and induced interactions

The dependence on the single-particle states of the pairing matrix elements of the Gogny force and of the bare low-momentum nucleon-nucleon potential vlow-k—designed so as to reproduce the low-energy observables avoiding the use of a repulsive core—is studied for a typical finite, superfluid nucleus (120Sn). It is found that the matrix elements of vlow-k follow closely those of vGogny on a wide range of energy values around the Fermi energy eF, those associated with vlow-k being less attractive. This result explains the fact that around eF the pairing gap DeltaGogny associated with the Gogny interaction (and with a density of single-particle levels corresponding to an effective k mass mk[approximate]0.7 m) is a factor of about 2 larger than Deltalow-k, being in agreement with Deltaexp=1.4 MeV. The exchange of low-lying collective surface vibrations among pairs of nucleons moving in time-reversal states gives rise to an induced pairing interaction vind peaked at eF. The interaction (vlow-k+vind) Zomega arising from the renormalization of the bare nucleon-nucleon potential and of the single-particle motion (omega-mass and quasiparticle strength Zomega) associated with the particle-vibration coupling mechanism, leads to a value of the pairing gap at the Fermi energy Deltaren that accounts for the experimental value. An important question that remains to be studied quantitatively is to what extent DeltaGogny, which depends on average parameters, and Deltaren, which explicitly depends on the parameters describing the (low-energy) nuclear structure, display or not a similar isotopic dependence and whether this dependence is borne out by the data.