Validity of a Local Approximation to the One-Pion-Exchange Potential for the Two-Nucleon System

The authors consider a nonlocal potential (nonlocal in coordinate space) which has been used to describe the exchange of one pion between two nucleons, and which when used in the first Born approximation, reproduces the relativistic $S$ matrix exactly to order ${g_{\pi }}^2$, the $\pi -N$ coupling strength squared. This nonlocal potential is compared with its lowest-order (in powers of $\frac{\mathbf{\nabla }}{m}$, where $m$ is the nucleon mass) local approximation. This comparison is made by inserting each of the potentials into the Lippmann-Schwinger (LS) equation and comparing the resulting phase parameters. The form factor of Ueda and Green is used to obtain the required convergence. Results are obtained which show that the local potential is in general not a reliable approximation to the nonlocal potential, even for low energies, when used in conjunction with the LS equation. As a by-product of this research, numerical methods are developed which are applicable to any nonlocal potential.