Pairing theory of the symmetry energy

As a model which displays a picture of the symmetry energy as an energy of rotation in isospace of a Cooper pair condensate, a Hamiltonian with a pairing force and an interaction of isospins is analyzed in the Hartree-Bogolyubov (HB) plus Random Phase (RPA) approximation. The HB energy minus Lagrangian multiplier terms is shown to be locally minimized by a product of neutron and proton Bardeen-Cooper-Schrieffer states. Nambu-Goldstone RPA solutions appear due to global gauge invariance and isobaric invariance. In an idealized case of infinitely many equidistant single-nucleon levels, the symmetry energy is composed of contributions from the single-nucleon and isospin interaction energies and the RPA correlation energy. The contribution of the latter is dominated by a neutron-proton Nambu-Goldstone solution, which makes the total symmetry energy nearly proportional to T(T+1). Observations reported from Skyrme force calculations are discussed in the light of these results. Calculations with deformed Woods-Saxon single-nucleon levels give results similar to those of the idealized case, whereas a somewhat different behavior is found with spherical Woods-Saxon levels. The calculations with Woods-Saxon single-nucleon levels reproduce surprisingly well the empirical symmetry energy.