High-Energy Collision Processes in Quantum Electrodynamics. III

We obtain explicitly the asymptotic behavior of the matrix elements at high energies for Delbrück scattering, i.e., the elastic scattering of a photon by a static Coulomb field via a virtual electron-positron pair. This is the simplest nontrivial two-body elastic scattering process in quantum electrodynamics besides those discussed in the preceding paper. The considerations are limited to the lowest order, $Z^2{e}^6$, for the matrix elements, and it is found that, to this order, ${\lim }_{\omega \to \infty }\frac{d\sigma }{\mathrm{dt}}$ exists and is nonzero for any fixed positive momentum transfer. This limiting value is expressed in terms of integrals, which are evaluated numerically. The behavior of these integrals is also studied in detail for the cases where the momentum transfer is either much larger or much smaller than the mass of the electron. Moreover, the scattered photon is significantly polarized in the scattering plane. None of the present results agree with the earlier ones of Bethe and Rohrlich using the impact-parameter approximation.