SOLUTION OF THE GAP EQUATION IN NEUTRON MATTER

The problem of solving the gap equation for S-wave pairing in pure neu tron matter is considered for the case that the pairing matrix element s V(p, p') are calculated directly from a realistic bare neutron-neutr on potential containing a strong short-range repulsion. The original g ap equation is replaced identically by a coupled set of equations: a n on-singular quasilinear integral equation for the dimensionless gap fu nction chi(P) defined by Delta(P) = Delta(F) chi(P) and a non-linear a lgebraic equation for the gap magnitude Delta(F) = Delta(PF) at the Fe rmi surface. This reformulation admits a robust and rapidly convergent iteration procedure for the determination of the gap function. The tr eatment may be extended to singlet or tripler pairing in non-zero angu lar momentum states. S-wave pairing is investigated numerically for th e Reid-soft-core interaction. Although the pairing matrix elements of this potential are everywhere positive, non-trivial solutions of the g ap equation are obtained on the range 0 < PF < P-c = 1.7496... fm(-1) of Fermi momenta, with the gap parameter Delta(F) reaching a maximum o f some 3 MeV near PF = 0.85 fm(-1). Numerical results are also provide d for the highly realistic Argonne nu(14) and nu(18) interactions. Wit hin the context of the new computational scheme, a condition for closu re of the gap is derived in terms of the first zero P0 of the gap func tion Delta(P). it is shown that Delta(F) vanishes exponentially not on ly in the low-density limit PF --> 0, but also as the Fermi momentum r ises and approaches the upper critical value p(c) specified by PF = P0 (PF), beyond which there exists no non-trivial solution of the gap equ ation. The numerical results for the function Delta(P) in neutron matt er display a remarkable universality of structure, visible especially in the stability of P0 under variation of density. Upon renormalizing the gap equation in terms of the vacuum S-wave scattering amplitude, t his behavior is seen to be a manifestation of the resonant nature of t he neutron-neutron interaction at low energy, which leads to a scatter ing amplitude of nearly separable form.