Summation of two-pion inserts to the sigma-meson propagator and nucleon forces

Each pion-pair insert into a sigma-meson propagator leads to a factorizable contribution, a function of the magnitude of the sigma-meson four-momentum. The sum of the resulting geometric series produces a modified propagator with a mass shift and a width which are determined by the physical mass and decay rate of the sigma meson. The modified propagator is used to calculate the contribution of all nucleon-nucleon scattering diagrams in which the nucleons can be separated by cutting only a sigma-meson line. This replaces the treatment of F. Partovi and E. L. Lomon in which the effect of the sigma-meson width was approximated by averaging the potential obtained from an unmodified propagator over the sigma-meson mass distribution. The new potential, when the general chiral constraint is imposed on the coupling constants, is small, in spite of large coupling strengths, due to cancellations between terms. It is similar to the previous approximate result and is less sensitive to the sigma-meson width. The size of the potential is reasonable for widths greater than 300 MeV. As before, the Weinberg-model prediction of a sigma meson which has a rho-meson mass and a width of 620 MeV leads to the best cancellation of nucleon-pair terms away from zero four-momentum transfer. The addition of this contribution to the theoretical potential of M. Hossein Partovi and E. L. Lomon tends to decrease the agreement with several realistic potentials. The size of the sigma-meson-exchange contribution is decreased if the sigma-meson width is increased; but neglecting exchange contributions a zero-isospin two-pion bound state also arises