Scattering amplitudes at high energies. III. An approach with Green's functions

The scattering amplitude at high energy is calculated in various approximations, explicitly incorporating both the single and double hard collisions. For the purpose of improving the large-angle behavior of the amplitude, approximations are introduced directly to the full Green's function rather than to the scattering function, thus avoiding the usual assumption of a straight-line trajectory. The previous study using the semiclassical Green's function gave very accurate amplitude up to a moderately large angle, and we have further examined its applicability to more general forms of potentials. Some of the practical difficulties in carrying out the amplitude integrals are pointed out, and the problem is simplified by introducing an angle-averaging procedure and also by an eikonal approximation to the semiclassical Green's function. These procedures are then tested using Gaussian and Yukawa forms of the potential; they are relatively simple to apply and the over-all accuracy is improved.