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