Relativistic eikonal expansion

Abstract

The generalized ladder series of Feynman diagrams for scattering of two particles by scalar-meson exchange is expanded, using functional methods, to obtain the relativistic eikonal approximation and the next two terms of an expansion about the eikonal limit. The established similarity between nonrelativistic and relativistic eikonal approximations is shown to persist, in part, to the higher-order terms in the relativistic eikonal expansion. The leading-order correction to the eikonal limit differs only kinematically from its nonrelativistic counterpart. In second order, there is again much similarity with nonrelativistic results; however, a part of the second-order eikonal correction explicitly depends on the relative time coordinate of the scattering particles. An approximate relativistic Schrodinger equation is found to reproduce the leading corrections to the eikonal limit by means of a simple kinematic generalization of the nonrelativistic potential theory results; however, the relativistic time effect cannot be readily incorporated , into a three-dimensional wave equation.

Coulomb corrections for quasielastic scattering: eikonal approximation

“…A good choice is R ′ = 0.5174R e . Furthermore, expression (23) has the advantage that it is possible to derive an analytic expression for the eikonal phase, which is convenient when the eikonal phase has to be calculated numerically in a computer program.…”

“…The eikonal phase turns out to be divergent for a Coulomb-like potential, but it is possible to regularize the eikonal phase by subtracting a screening potential ∼ (r 2 + a 2 ) −1/2 with a ≫ R from (23), such that the potential falls off like r −2 for large r. The divergence can then be absorbed in a constant divergent phase ∼ log(a) without physical significance, when the limit a → ∞ is taken. It is quite instructive to calculate the eikonal phase for the simple screened potential [29] …”

“…The expressions necessary for the calculation of the eikonal phase for the potential (23) are given in the appendix. Potential V used in our calculations, potential V s generated by a homogeneously charged sphere with radius R e and potential generated by a corresponding Woods-Saxon charge distribution.…”

“…Various approximate treatments have been proposed in the past for the treatment of CC [10,12,13,14,15,16,17,18,19], and there is an extensive literature on the eikonal approximation [20,21,22,23,24]. In particular it has been shown that at lowest order, an expansion of the electron wave function in αZ leads to the well known effective momentum approximation (EMA) [9], which is explained in Sect.…”

Focusing of high-energy particles in the electrostatic field of a homogeneously charged sphere and the effective momentum approximation

“…As a consequence, various approximate methods have been proposed in the past for the treatment of Coulomb distortions [7,8,9,10,11,12,13,14,15], and there is an extensive literature on the so-called eikonal approximation [16,17,18,19,20,21,22,23,24].…”