A Proposal to Measure Photon-Photon Scattering

We discuss a possibility to measure the photon-photon scattering cross section at low energy in a theoretical standpoint. The cross section of photon-photon scattering at low energy can be written as $\displaystyle{{d\sigma\over d\Omega} \simeq {\alpha^4\over (12\pi)^2 \omega^2} (3+2\cos^2\theta +\cos^4\theta)}$ with $\omega$ the energy of photon. The magnitude of the cross section at $\omega \simeq 1$ eV should be $10^{37}$ times larger than the prediction of Heisenberg and Euler who calculated the photon scattering by the classical picture of field theory. Due to a difficulty of the initial condition of photon-photon reaction process, we propose to first measure $\gamma +\gamma \rightarrow e^++e^-$ reaction at a few MeV before measuring $\gamma +\gamma \rightarrow \gamma +\gamma$ elastic scattering.