Effects of pion-fold-pion diagrams in the energy-independent nucleon-nucleon potential

Abstract

Based on a T-matrix equivalence theory, an energy-independent or locally energy-dependent nucléon-nucléon potential V NN derived from meson exchanges is studied. The potential, given as a series expansion of folded diagrams, is independent of the asymptotic energy of the scattering nucleons. It is, however, locally energy dependent in the sense that its matrix elements 〈 a| V NN| b〉 depend on the energies associated with its bra and ket states a and b. Our formulation makes use of right-hand-side on-shell T-matrix equivalence of the field-theoretical and potential descriptions when limited to the space of neutrons and protons only. This preserves not only scattering (e.g. phase shifts, projections of wave functions) but also bound-state properties. The matrix elements of V were calculated for two potential models, one based on one-pion exchange (OPEP) and the other on one-boson exchange (OBEP) using {π, ρ, σ, ω, δ, η }. Three types of phase-shift calculations have been carried out to study the viability of constructing an energy-independent potential using the folded-diagram expansion: (A) NN phase shifts for an energy-dependent OPEP and OBEP. For the OBEP we used parameters adjusted to fit experimental data. (B) The same phase shifts for the energy-independent case for both OPEP and OBEP. (C) Repetition of (B) with effects of the two-pion folded diagrams included. Our results show two important points: (i) folded diagrams are of essential importance, and (ii) the first-order folded diagrams contain the dominant effect and the neglect of terms with more than two folds can be regarded as a good approximation. The effects of folded diagrams are large especially for low partial waves and high energies. For high partial waves ( J greater than 2) the folded terms are negligible, and the phase shifts given by (A), (B) and (C) practically coincide.