High-Energy Expansions of Scattering Amplitudes

The partial-wave series is converted without approximation to a Fourier-Bessel expansion based on a new (infinite series) expansion for the Legendre function. The direct connection between the Fourier-Bessel phase shift and the partial-wave interpolating phase shift is established as an infinite series in powers of $K^{-2\mathrm{}}$ ($K=\mathrm{wave}\mathrm{number}$). The series contains the Glauber eikonal approximation as a leading term and reproduces the results of an eikonal expansion about the Glauber propagator. Corrections to the eikonal approximation are developed and rules are given for an unambiguous interpretation of the eikonal expansion. The relativistic eikonal expansion is discussed for forward and backward scattering without small-angle approximations.