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 $\ensuremath{\sigma}$ meson and any number of pion-pair inserts on the $\ensuremath{\sigma}$-meson propagator and vertices. The reduction of the Bethe-Salpeter kernel to a potential follows the method of Partovi and Lomon. The $\ensuremath{\sigma}$-pion and $\ensuremath{\sigma}$-nucleon couplings are restricted by the Adler chiral condition at zero four-momentum transfer, renormalized. The $\ensuremath{\sigma}$-meson mass is taken from experimental evidence to be nearly ${M}_{\ensuremath{\rho}}$. Various $\ensuremath{\sigma}$-meson widths are considered. Widths greater than 300 MeV predict small $\ensuremath{\sigma}$-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 $\ensuremath{\sigma}$-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, $\ensuremath{\eta}\ensuremath{-}$, $\ensuremath{\rho}\ensuremath{-}$, and $\ensuremath{\omega}\ensuremath{-}$ 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 $\ensuremath{\sigma}$-meson widths near 620 MeV and 660 Mev, respectively.