Pairing Matrix Elements and Pairing Gaps with Bare, Effective and Induced Interactions

Abstract

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 $v_{low-k}$ is studied in the semiclassical approximation for the case of a typical finite, superfluid nucleus (${}^{120}$Sn). 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 ${\Delta }_{Gogny}$ associated with the Gogny interaction (and with a density of single-particle levels corresponding to an effective $k$-mass $m_k\approx 0.7m$) is a factor of about 2 larger than ${\Delta }_{low-k}$,being in agreement with ${\Delta }_{exp}$= 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 $(v_{low-k}+v_{ind})Z_{\omega }$ arising from the renormalization of the bare nucleon-nucleon potential and of the single-particle motion ($\omega -$mass and quasiparticle strength $Z_{\omega }$) due to the particle-vibration coupling leads to a value of the pairing gap at the Fermi energy ${\Delta }_{ren}$ which accounts for the experimental value.