Ring-diagram summations and the self-energy of the homogeneous electron gas at its weak-correlation limit

The small-r(s) asymptotics of the self-energy Sigma(k,omega) of the homogeneous electron gas (HEG) is studied in terms of the Feynman diagrams involving the noninteracting one-body Green's function G(0) and the static bare Coulomb repulsion v(0). The lowest-order approximation to Sigma(k,omega) is given by the product of G(0) and v(0). The nature of the proper ring-diagram summation (equivalent to the random-phase approximation) for Sigma(k,omega) that affords the correct small-r(s) single behavior of r(s)(2) ln r(s) is investigated. Reexamination of ring-diagram summations for several properties of the HEG proves in a rigorous manner that the product G(0)v(r), where v(r) is the ring-diagram-summed dynamically screened repulsion, yields the correct lowest-order asymptotics, whereas G(r)v(0), where G(r) is the ring-diagram-summed Green's function, contributes only to higher-order terms.