Field-Theoretical Nucleon-Nucleon Potential

TLDR: In this article, the relativistic two-body equation of Bethe and Salpeter is reduced using a generalization of the Blankenbecler-Sugar method, which is shown to be identical with the nonrelativistic Lippmann-Schwinger equation upon a unitarity-preserving identification of the amplitudes.

Abstract: A formulation is presented for the derivation of a Schr\"odinger-equation potential from a field-theoretical model. The relativistic two-body equation of Bethe and Salpeter is reduced using a generalization of the Blankenbecler-Sugar method. The resulting equation is shown to be identical with the nonrelativistic Lippmann-Schwinger equation upon a unitarity-preserving identification of the amplitudes. An equivalent potential is thereby defined and expressed as a solution of an integral equation. The second- and fourth-order potentials are calculated, and their energy dependence and nonlocality are studied. An approximation scheme is developed for expanding the configuration-space potentials in the powers of the momentum operator. Terms up to and including the first power are retained, giving rise to a potential composed of central, spin-orbit, tensor, and spin-spin parts. The contributions of the meson resonances $\ensuremath{\eta}$, $\ensuremath{\rho}$, and $\ensuremath{\omega}$ are included to second order. The complete potential is numerically calculated using masses and coupling parameters taken from meson experiments; no parameter of the potential is searched upon. The resulting potential is remarkably similar to that of Hamada and Johnston (outside half a pion Compton wavelength), particularly for the parts that are relatively well determined by nucleon-nucleon scattering data. Further extensions of the program, including the treatment of the nucleon resonances and pair suppression, are discussed, and an outline of such extensions is given.