Self-energy, momentum distribution, and effective masses of a dilute Fermi gas

The dependence upon momentum $k$ and upon energy $E$ of the self-energy of a dilute Fermi gas is studied up to terms of order ${\left(k_{F}c\right)}^2$, where $k_F$ denotes the Fermi momentum and $c$ is the positive scattering length. Algebraic expressions are derived for the imaginary part $W(k;E)\mathrm{}$ of the self-energy in the whole ($k$,$E$) plane. They are compared with a conjecture recently made by Orland and Schaeffer in their analysis of single-particle states in nuclei. The contributions of core polarization and of ground state correlations to the real part $V(k;E)\mathrm{}$ of the self-energy are calculated with the help of subtracted dispersion relations which connect them with $W(k;E^{\prime })$. Algebraic expressions are derived for the momentum distribution in the correlated ground state. It is shown that the effective mass of a quasiparticle with momentum $k$ is equal to the bare particle mass at $k=0\mathrm{}$ and reaches a local maximum for $k$ close to $k_F$. This maximum is ascribed to the dependence of $V(k;E)\mathrm{}$ upon $E$, which is described in terms of an $E$ mass. We compute the contributions of core polarization and of ground state correlations to this $E$ mass. The dependence of $V(k;E)\mathrm{}$ upon $k$ reflects the nonlocality of the self-energy. It is characterized by a $k$ mass that we also calculate. These results shed light on some nuclear matter properties and on the meaningfulness and limitation of nuclear matter calculations that have recently been performed in the framework of the Brueckner-Hartree-Fock approximation.