Comparison of Approximate Methods for Multiple Scattering in High-Energy Collisions

We compare the efficacy of several approximate methods for dealing with multiple-scattering processes by applying these methods to a simple, soluble model. The model consists of the scattering of a particle by a pair of like particles with motion of all particles constrained to one dimension. Two cases are considered: (1) the target particles have a fixed separation and (2) the target particles are in a deuteron-like bound state with each other. The interaction of the incident particle with each target particle is the same and is chosen to be a zero-range potential. The potential binding the two target particles in case 2 is equal to this same zero-range interaction. The methods compared are the Born approximation for the $T$ matrix, the impulse approximation for the $T$ matrix, the WKB approximation, the Glauber approximation, second-order multiple-scattering theory for the $T$ matrix, Born approximation for the $K$ matrix, impulse approximation for the $K$ matrix, and second-order multiple-scattering theory for the $K$ matrix. For the fixed-separation case the $K$-matrix formalisms work much better than the $T$-matrix formalisms including Glauber theory. Second-order multiple-scattering theory is superior to Glauber theory for fixed separation. For case 2 the aforementioned approximate methods are used in conjunction with the adiabatic approximation. Comparison of the calculated transmission probabilities with the exact transmission probability shows the Glauber theory doing much better than any of the other approximate methods. The success of the Glauber theory results from its suppression of a second-order term in the multiple-scattering expansion which is inadequately damped by the adiabatic approximation.