High-Energy Approximations in Potential Scattering Theory

The relationship between the eikonal approximation of Glauber and the impact-parameter approximation of Blankenbecler and Goldberger is clarified. This is done by comparing the approximate forms for the scattering amplitude for nonrelativistic scattering by a potential with the Born expansion, which at high energy assumes the natural form of a series of terms in inverse powers of the center-of-mass momentum p. It is found that both approximations reproduce the first Born term and, to order 1/p, the on-energy-shell part of the second Born term, for all momentum transfers. Furthermore, the Glauber approximation correctly gives the leading contribution (in powers of 1/p) to the on-energy-shell part of every term in the Born series, whereas the Blankenbecler-Goldberger approximation fails to do this for Born terms beyond the second. However, since the comparison to the Born series shows that both approximations neglect contributions of order 1p², neither approximation, in the simple forms commonly used, is correct beyond terms of order 1/p.