Bethe-Salpeter Solution for Nucleon-Nucleon Scattering with Pion Exchange in theS01andP03States

The Bethe-Salpeter equation is solved for nucleon-nucleon scattering in the ladder approximation with pion exchange for the ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ and ${}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}$ states with no approximations beyond use of a finite mesh. It is found that solutions exist without the need for any cutoff so long as the coupling constant $\frac{g^2}{4\pi }$ is between about -4 and +7 for the ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ state and for $\frac{g^2}{4\pi }$ less than about +4 for the ${}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}$ state. These results are in qualitative agreement with the predictions of Mandelstam. It is found that the Padé approximant as applied to the coefficients of the iteration solution provides an efficient alternative to the method of matrix inversion for solving the equations for a given mesh.