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 v_low-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 v_low-k follow closely those of v_Gogny on a wide range of energy values around the Fermi energy e_F, those associated with v_low-k being less attractive. This result explains the fact that around e_F the pairing gap Δ_Gogny associated with the Gogny interaction (and with a density of single-particle levels corresponding to an effective k mass m_k≈0.7 m) is a factor of about 2 larger than Δ_low-k, being in agreement with Δ_exp=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 v_ind peaked at e_F. The interaction (v_low-k+v_ind) Z_ω arising from the renormalization of the bare nucleon-nucleon potential and of the single-particle motion (ω-mass and quasiparticle strength Z_ω) associated with the particle-vibration coupling mechanism, leads to a value of the pairing gap at the Fermi energy Δ_ren that accounts for the experimental value. An important question that remains to be studied quantitatively is to what extent Δ_Gogny, which depends on average parameters, and Δ_ren, 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.