Summation of particle-particle and particle-hole ring diagrams in binding energy calculations and Lipkin models

Abstract

In this paper we study methods for calculating the particle-particle (pp) and particle-hole (ph) ring diagrams in the linked diagram expansion of the ground-state energy shift Δ E 0 for many-body systems. Analytic expressions for summing up the pp ring diagrams to all orders are derived, given as an integral involving G pp( ω, λ) ν, where G pp is the pp Green function, υ the interaction hamiltonian (taken to be two-body) and λ a strength parameter to be integrated over from 0 to 1. Similar expressions for summing up the ph ring diagrams are also derived, their integral form involving G ph (ω, λ) ν, where G ph is the ph Green function and ν the particle-hole interaction. T energy variable ω is to be integrated over from −∞ to +∞. Two methods for carrying out the ω-integration are suggested. One is the multipole expansion method, and the other is the transition amplitude method where the Green functions G ppand G ph are calculated using RPA equations from which the physical transition amplitudes can be obtained. Our methods for summing up the pp and ph ring diagrams to all orders are applied to several Lipkin-model many-body problems. The effect of using Hartree-Fock self-consistent single-particle wave functions on the contribution of high-order ring diagrams to Δ E 0 is studied and found to be very important. Possible applications of our methods to actual nuclear many-body problems are discussed.