Off-the-energy-shell properties of the mass operator and spectral functions in nuclear matter

Abstract

We investigate the off-the-energy-shell properties of the mass operator M( k; ω) = V( k; ω) + iW( k; ω), i.e., its dependence upon the nucleon momentum k and upon the nucleon frequency ω, separately. Particular attention is paid to the dispersion relation which connects its real and imaginary parts. We limit ourselves to the first two terms of the hole-line expansion of the mass operator, namely to the Brueckner-Hartree-Fock field M 1( k; ω) and to the second-order “rearrangement” term M 2( k; ω). Most previous works only dealt with “on-shell values” obtained by setting ω equal to the root e( k) of an energy-momentum relation, or equivalently by setting k equal to the root k( e) of this energy-momentum relation. We use as input a finite-rank representation of the realistic Argonne v 14 nucleon-nucleon interaction. The Fermi momentum is set equal to 1.36 fm −1. For momenta larger than the Fermi momentum, the calculated k-dependence of the on-shell depth V 1( k; e( k)) can be approximated by a gaussian. The corresponding nonlocality range is close to that assumed by Perey and Buck in their phenomenological analysis of scattering cross sections; it is somewhat smaller than that associated with the k-dependence of the off-shell potential V 1( k; ω) for fixed ω. The calculated ω-dependence of V 2( k; ω) is in excellent agreement with the dispersion relation which connects V 2( k; ω) to the values of W 2( k; ω′) for all ω′< e( k 1). The dispersion relation between V 1( k; ω) and W 1( k; ω′) is also investigated; in that case, caution must be exercised because the values of W 1( k; ω′) for ω′ larger than 500 MeV still play a sizeable role, and also because the dispersion relation involves a large ω-independent “background”; it is proved that the latter is equal to the Hartree-Fock potential. More generally, the dispersion relation between the real and imaginary parts of the exact mass operator involves an ω-independent background for which we derive a closede xpression analogous to but different from the Hartree-Fock potential. The ω-dependence of the spectral function S( k; ω) is calculated for two typical values of k, namely 3 4 k F and 5 4 k F . Its integral over all values of ω differs from unity by only a few percent, which provides an estimate of the reliability of our approximation scheme. We study the dependence upon E A + 1 ∗ of the integrated “particle” strength located below the excitation energy E A + 1 ∗ in the system formed by adding a nucleon with momentum k to the nuclear-matter ground state. We also calculated the dependence upon E A − 1 ∗ of the integrated “hole” strength located below the excitation energy E A − 1 ∗ in the system formed by taking out a nucleon with momentum k from the nuclear-matter ground state. In the limit E A − 1 ∗ → ∞ , this integrated hole strength yields an approximation of the occupation probability of the momentum k in the ground state. We compare this result with estimates obtained from other approximate expressions which involve the partial derivative of V 1( k; ω) with respect to ω. We also evaluate the “mean removal energies” and compare them to the “quasiparticle energies,” i.e., to the energies at which the spectral function presents a maximum.