Pairing with polarization effects in low-density neutron matter

The problem of the influence of the medium on the effective pairing interaction in nuclear matter is a long standing one that still awaits satisfactory solution. This is true even for the simplest case of pairing in the S0 channel in pure neutron matter, to which this article is restricted. A quantitative control on this issue would be very useful in particular for the understanding of neutron star physics @1#, where several physical phenomena ~cooling, glitches! are thought to depend very sensitively on the size and the density dependence of the gap. In several publications @2–4# the gap equation is solved in the simplest ~BCS! approximation @5–10#, namely using the bare neutron-neutron potential as interaction kernel. With a realistic nucleon-nucleon potential, adapted to the scattering phase shifts, one obtains typically a maximum of the gap D(kF)'3 MeV at a density corresponding to kF'0.85 fm. However, the use of the bare potential completely disregards the influence of the surrounding neutron medium and some authors have attempted to go beyond this level by considering certain additional subsets of diagrams in the interaction kernel @3,11–15#. Unfortunately, doing so the interaction becomes rather complex, and therefore always certain approximations ~phase space averages, weak-coupling approximation, . . . ) have to be performed in order to arrive at a numerically feasible level. It is well known, however, that the solution of the gap equation depends exponentially on the strength of the interaction, so that any kind of approximation has to be introduced with great care. Also the choice of a particular subseries of graphs has to be considered in this light. Nevertheless, the previous works agree in predicting an important suppression of the pairing gap. However, the precise quantitative level of this suppression as well as its density dependence vary substantially with the different approaches and must be considered unknown for the time being. An overview of the previous results can be found in Ref. @3#, for example. This is the motivation to attempt in this article to face the problem from a different viewpoint, namely a systematic, perturbative treatment valid at low density. The number of further approximations at this level should be kept to a strict minimum. In this article we present the first step within such an approach. More precisely, we will extend the lowest-order interaction kernel by the complete set of diagrams of second order in the interaction and containing one hole line.