High-Energy Delbrück Scattering Close to the Forward Direction

Delbrück 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 $\Delta$, namely, $m$ and $\frac{m^2}{\omega }$, where $m$ is the mass of the electron, and $\omega$ is the photon energy. When $\Delta$ is much larger than the smaller scale $\frac{m^2}{\omega }$, the impact factor representation holds at high energies. The impact picture is here extended to give also the high-energy behavior of the Delbrück scattering amplitude when $\Delta$ is comparable to $\frac{m^2}{\omega }$. The result can be expressed in terms of generalized hypergeometric functions, which reproduces the known result in the forward direction when $\Delta$ is set equal to zero, and also joins smoothly to the impact factor representation when $\Delta$ is much larger than $\frac{m^2}{\omega }$. In the present analysis, the fine-structure constant $\alpha$ is assumed to be small, but not $Z\alpha$. In other words, all terms of the form $\alpha {\left(Z\alpha \right)}^n$ in the amplitude are taken into account. It is also shown that the result for $\Delta \ll m$ is independent of the mass of the target, and hence is in particular applicable to Compton scattering by an electron.