Solution of the gap equation in neutron matter

Abstract

Abstract The problem of solving the gap equation for S-wave pairing in pure neutron matter is considered for the case that the pairing matrix elements V ( p , p ′) are calculated directly from a realistic bare neutron-neutron potential containing a strong short-range repulsion. The original gap equation is replaced identically by a coupled set of equations: a non-singular quasilinear integral equation for the dimensionless gap function χ ( p ) defined by Δ ( p ) = Δ F χ ( p ) and a non-linear algebraic equation for the gap magnitude Δ F = Δ ( p F ) at the Fermi surface. This reformulation admits a robust and rapidly convergent iteration procedure for the determination of the gap function. The treatment may be extended to singlet or triplet pairing in non-zero angular momentum states. S-wave pairing is investigated numerically for the Reid-soft-core interaction. Although the pairing matrix elements of this potential are everywhere positive, non-trivial solutions of the gap equation are obtained on the range 0 p F p c = 1.7496… fm −1 of Fermi momenta, with the gap parameter Δ F reaching a maximum of some 3 MeV near p F = 0.85 fm −1 . Numerical results are also provided for the highly realistic Argonne υ 14 and υ 18 interactions. Within the context of the new computational scheme, a condition for closure of the gap is derived in terms of the first zero p 0 of the gap function Δ ( p ). It is shown that Δ F vanishes exponentially not only in the low-density limit p F → 0, but also as the Fermi momentum rises and approaches the upper critical value p c specified by p F = p 0 ( p F ), beyond which there exists no non-trivial solution of the gap equation. The numerical results for the function Δ ( p ) in neutron matter display a remarkable universality of structure, visible especially in the stability of p 0 under variation of density. Upon renormalizing the gap equation in terms of the vacuum S-wave scattering amplitude, this behavior is seen to be a manifestation of the resonant nature of the neutron-neutron interaction at low energy, which leads to a scattering amplitude of nearly separable form.