σ -Exchange Potential for Nucleon-Nucleon Scattering

A nucleon-nucleon interaction potential is derived which includes all exchanges between a pair of nucleons involving a single σ meson and any number of pion-pair inserts on the σ-meson propagator and vertices. The reduction of the Bethe-Salpeter kernel to a potential follows the method of Partovi and Lomon. The σ-pion and σ-nucleon couplings are restricted by the Adler chiral condition at zero four-momentum transfer, renormalized. The σ-meson mass is taken from experimental evidence to be nearly M_ρ. Various σ-meson widths are considered. Widths greater than 300 MeV predict small σ-meson contributions to the potential due to persistent cancellations between graphs. In particular near the width of about 610 MeV predicted by Weinberg's asymptotic restrictions the σ-potential contribution is a minimum and nearly cancels the familiar one-plus-two nucleon-pair contribution to the potential. The corrections to the previously obtained single-pion, η-, ρ-, and ω- meson and two-pion potentials are such as to increase the resemblance to Hamada-Johnston and Reid potentials or increase the resemblance to the Feshbach-Lomon potential for σ-meson widths near 620 MeV and 660 Mev, respectively.