The BCS pairing gap in the on-shell limit of the Similarity Renormalization Group

Abstract

The pairing gap plays a fundamental role in the nuclear many-body problem and many large scale and accurate mass formula fits suggest the smooth nuclear mass dependence $\Delta \sim 6(1)A^{-1/3}\mathrm{MeV}$ in the liquid drop model which lacks a theoretical motivation. Within the BCS theory we analyze the impact of phase equivalent interactions on the pairing gap for a translational invariant many-fermion system such as nuclear and neutron matter. To that end we use explicitly the Similarity Renormalization Group (SRG) transformations. We show that in the on-shell and continuum limits the pairing gap vanishes. For finite size systems the pairing gap can be computed directly from the scattering phase-shifts by the formula $${\Delta }_{nn}(p_F)=\Delta {ϵ}_F{\delta }_{nn}^{{}^1{S}_0}(p_F)/\pi ,$$ where $p_F$ is the Fermi momentum and $\Delta {ϵ}_F$ the level spacing at the Fermi energy which for the harmonic oscillator shell model becomes $\Delta {ϵ}_F=\hslash \omega \sim 41A^{-1/3}\mathrm{MeV}$, so that $${\Delta }_{nn}(p_F)\sim 4A^{-1/3}\mathrm{MeV}.$$ The comparison with double differences from binding energies of stable nuclei is satisfactory and the discrepancy with the large scale analysis may be attributed to the lack of three-body forces. Nevertheless, the on-shell two-body interaction provides a basis for the $c{A}^{-1/3}$ dependency and accounts for 75\% of the coefficient $c$.

Comments: 10 pages, 4 figures. Major changes, References added