Photon-Photon Scattering close to the Forward Direction

Abstract

The scattering amplitude for photon-photon elastic scattering in quantum electrodynamics is explicitly obtained for the case where the energy is very large while the momentum transfer is small compared with the rest energy of the electron (divided by $c$). This amplitude has no infrared divergence and, at high energies, the fourth- and sixth-order contributions are negligible. The explicit answer for the eighth order is remarkably simple. In terms of helicity states, it is found that the amplitudes $\mathrm{RR}\to \mathrm{RR}$ and $\mathrm{RL}\to \mathrm{RL}$ are the same, while the amplitude $\mathrm{RR}\to \mathrm{LL}$ is different but of the same order of magnitude. By the optical theorem, the total cross section for $\gamma +\gamma \to e^{-}+e^{-}+e^{+}+e^{+}$ is found to be 6.5 μb at high energies.