Higher-order terms in the eikonal expansion of theTmatrix for potential scattering

Abstract

In this paper we evaluate some higher-order terms in the eikonal expansion of the $T$ matrix for potential scattering. We calculate eikonal correction terms which are of second order in potential strength ${V}_{0}$ through seventh order in inverse momentum ${K}^{\ensuremath{-}1}$, which leads us to conjecture two plausible formulas for general eikonal correction terms of $O({K}^{\ensuremath{-}2n}{{V}_{0}}^{2})$ and of $O({K}^{\ensuremath{-}2n\ensuremath{-}1}{{V}_{0}}^{2})$. This may enable us to obtain an eikonal expansion of the second Born amplitude for any type of potential. We further evaluate eikonal correction terms which are of third order in potential strength and fourth and fifth order in ${K}^{\ensuremath{-}}$. With the help of eikonal corrections evaluated in this paper, a general formula based on a WKB phase function, and a relation existing between the amplitude term and the eikonal phase of the impact-parameter representation of the scattering $T$ matrix, we are able to determine the $T$ matrix completely through fourth order in inverse momentum. We may also note that the formula for the eikonal correction term of $O({K}^{\ensuremath{-}2n\ensuremath{-}1}{{V}_{0}}^{2})$ when generalized in a suitable manner surprisingly yields the expression for the explicitly evaluated eikonal correction term of $O({K}^{\ensuremath{-}5}{{V}_{0}}^{3})$. It is further shown in this paper that the previously mentioned conjectured formula for the general eikonal phase correction term of $O({K}^{\ensuremath{-}2n\ensuremath{-}1}{{V}_{0}}^{2})$ is somewhat supported by the form of the corresponding correction term to the usual WKB phase-shift result. All the higher-order eikonal correction terms evaluated in this paper vanish for the Coulomb interaction.