Two-body Dirac equations for nucleon-nucleon scattering

Abstract

We investigate the nucleon-nucleon interaction by using the meson exchange model and the two-body Dirac equations of constraint dynamics. This approach to the two-body problem has been successfully tested for QED and QCD relativistic bound states. An important question we wish to address is whether or not the two-body nucleon-nucleon scattering problem can be reasonably described in this approach as well. This test involves a number of related problems. First we must reduce our two-body Dirac equations exactly to a Schr\"odinger-like equation in such a way that allows us to use techniques to solve them already developed for Schr\"odinger-like systems in nonrelativistic quantum mechanics. Related to this, we present a new derivation of Calogero's variable phase shift differential equation for coupled Schr\"odinger-like equations. Then we determine if the use of nine meson exchanges in our equations gives a reasonable fit to the experimental scattering phase shifts for $n\ensuremath{-}p$ scattering. The data involve seven angular momentum states including the singlet states ${}^{1}{S}_{0},$ ${}^{1}{P}_{1},$ ${}^{1}{D}_{2}$ and the triplet states ${}^{3}{P}_{0},$ ${}^{3}{P}_{1},$ ${}^{3}{S}_{1},$ ${}^{3}{D}_{1}.$ Two models that we have tested give us a fairly good fit. The parameters obtained by fitting the $n\ensuremath{-}p$ experimental scattering phase shift give a fairly good prediction for most of the $p\ensuremath{-}p$ experimental scattering phase shifts examined (for the singlet states ${}^{1}{S}_{0},$ ${}^{1}{D}_{2}$ and triplet states ${}^{3}{P}_{0},$ ${}^{3}{P}_{1}).$ Thus the two-body Dirac equations of constraint dynamics present us with a fit that encourages the exploration of a more realistic model. We outline generalizations of the meson exchange model for invariant potentials that may possibly improve the fit.