On Light-by-Light Interaction in QED

In light of the recent observation of light-by-light interaction at the LHC, we consider the possible two photon interaction in the QED vacuum when it is exposed to an intense electromagnetic field \(E_{\gamma }\) of counter-propagating charged particles. In particular, we investigate the photon-photon scattering \(\gamma \gamma \rightarrow X\longrightarrow \gamma \gamma \) through the virtual electron-positron paires \(X\equiv e^{-}e^{+}\) of the QED vacuum medium with induced deviated refraction index \(\eta _{vac} ^{QED}\left( E_{\gamma }\right) \simeq 1\pm \xi _{\gamma }E_{\gamma } ^{2}/m_{e^{\pm }}^{2}\). Then, according to the known and reported data we bound the corresponding photonic scattering cross section \(\sigma _{\gamma \gamma }\lesssim 10^{-34}cm^{2}\).