Non-Locality in the Second Order Potential

Abstract

The non-local properties of the second order two nucleon interaction are investigated using ps-ps meson theory and the Tamm-Dancoff approach. None of the customary transformations are applied to the potential or wave function, so the second order potential derived is both momentum dependent and energy dependent. The potential is hermitian, however, so that the two nucleon component of the wave function can be normalized in the usual manner, conserving probability. The size of the non-local corrections on scattering states is estimated as a function of energy and as a function of the internucleon separation distance. The Born approximation is used to show that the lowest order velocity correction accounts for almost all of the non-local corrections at 300 Mev laboratory energy where it decreases the static scattering amplitude by about 25 per cent. The energy dependence of the potential is instrumental for the Born approximation scattering amplitude being the same as given by lowest order S-matrix theory. The non-local corrections are estimated as a function of internucleon distance by defining an “effective” scattering potential which disguises most of the velocity terms as a static fourth order potential. Velocity corrections are found to be very large in triplet states, dominating over the static second order potential at about 1.5 ∼3.0 f. Comparison with potentials of others where various transformations have been employed is discussed. There is no spin orbit component in the second order ps-ps potential, but the corresponding potential derived from scalar meson theory does have such a component. The latter calculation suggests that a spin orbit potential should really be regarded as a quadratic momentum dependence and therefore does not have an apriori preference over other admissible quadratic dependences.