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

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 $\Delta \sim 6(1)~ A^{-1/3}~{\rm MeV}$ 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 $$\Delta_{nn} (p_F) = \Delta \epsilon_F ~ \delta^{^1S_0}_{nn}(p_F) /\pi ~ ,$$ where $p_F$ is the Fermi momentum and $\Delta \epsilon_F$ the level spacing at the Fermi energy which for the harmonic oscillator shell model becomes $\Delta \epsilon_F= \hbar \omega \sim 41 ~ A^{-1/3}~{\rm MeV}$, so that $$\Delta_{nn} (p_F) \sim 4 ~ A^{-1/3}~{\rm MeV} ~ .$$ 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 $c$.