High-Energy Delbrück Scattering Close to the Forward Direction

Delbr\"uck scattering is the elastic scattering of a photon by a static Coulomb field via electron-positron pair creation. At high energies, there are two natural scales for the momentum transfer Δ, namely, m and m²ω, where m is the mass of the electron, and ω is the photon energy. When Δ is much larger than the smaller scale m²ω, the impact factor representation holds at high energies. The impact picture is here extended to give also the high-energy behavior of the Delbr\"uck scattering amplitude when Δ is comparable to m²ω. The result can be expressed in terms of generalized hypergeometric functions, which reproduces the known result in the forward direction when Δ is set equal to zero, and also joins smoothly to the impact factor representation when Δ is much larger than m²ω. In the present analysis, the fine-structure constant α is assumed to be small, but not Zα. In other words, all terms of the form α(Zα)ⁿ in the amplitude are taken into account. It is also shown that the result for Δm is independent of the mass of the target, and hence is in particular applicable to Compton scattering by an electron.