Bare vs effective pairing forces: A microscopic finite-range interaction for Hartree-Fock-Bogolyubov calculations in coordinate space

Abstract

We propose a microscopic effective interaction to treat pairing correlations in the $^{1}S_{0}$ channel. It is introduced by recasting the gap equation written in terms of the bare force into a fully equivalent pairing problem. Within this approach, the proposed interaction reproduces the pairing properties provided by the realistic AV18 force very accurately. Written in the canonical basis of the actual Bogolyubov transformation, the force takes the form of an off-shell in-medium two-body matrix in the superfluid phase multiplied by a BCS occupation number $2{\ensuremath{\rho}}_{m}$. This interaction is finite-ranged, nonlocal, total-momentum dependent, and density dependent. The factor $2{\ensuremath{\rho}}_{m}$ emerging from the recast of the gap equation provides a natural cutoff and makes zero-range approximations of the effective vertex meaningful. Performing such an approximation, the roles of the range and of the density dependence of the interaction can be disentangled. The isoscalar and isovector density dependences derived ab initio provide the pairing force with a strong predictive power when extrapolated toward the drip lines. Although finite ranged and nonlocal, the proposed interaction makes Hartree-Fock-Bogolyubov calculations of finite nuclei in coordinate space tractable. Through the two-basis method, its computational cost is of the same order as for a zero-range force.