Long-range behavior of the optical potential for the elastic scattering of charged composite particles

Abstract

The asymptotic behavior of the optical potential, describing elastic scattering of a charged particle $α$ off a bound state of two charged, or one charged and one neutral, particles at small momentum transfer $Δ_α$ or equivalently at large intercluster distance $ρ_α$, is investigated within the framework of the exact three-body theory. For the three-charged-particle Green function that occurs in the exact expression for the optical potential, a recently derived expression, which is appropriate for the asymptotic region under consideration, is used. We find that for arbitrary values of the energy parameter the non-static part of the optical potential behaves for $Δ_α \rightarrow 0$ as $C_{1}Δ_α + o\,(Δ_α)$. From this we derive for the Fourier transform of its on-shell restriction for $ρ_α \rightarrow \infty$ the behavior $-a/2ρ_α^4 + o\,(1/ρ_α^4)$, i.e., dipole or quadrupole terms do not occur in the coordinate-space asymptotics. This result corroborates the standard one, which is obtained by perturbative methods. The general, energy-dependent expression for the dynamic polarisability $C_{1}$ is derived; on the energy shell it reduces to the conventional polarisability $a$ which is independent of the energy. We emphasize that the present derivation is {\em non-perturbative}, i.e., it does not make use of adiabatic or similar approximations, and is valid for energies {\em below as well as above the three-body dissociation threshold}.