Regge-pole description of rainbow scattering by means of the phase-integral method

Abstract

In the present paper we study the effects in the rainbow differential cross sections of using higher-order phase-integral residues in the pole sum of the scattering amplitude. We have recently found that already in the first-order approximation the uniform residue formula, derived by Fr\"oman and Fr\"oman, provides an important means for an accurate calculation of the rainbow differential cross section where reliable numerical quantum methods are not available. From the present investigation it is clear that the use of higher-order approximations is not the crucial step to further reveal the rainbow structures in the cross section. For that purpose, it is instead essential to increase the purely numerical precision (in any order) of the uniform phase-integral residue calculations.