Effective Lagrangians and low energy photon-photon scattering

We investigate the behavior of the photon-photon scattering near the threshold $\omega = m_e$, by comparing the exact result with the approximation obtained using the Euler-Heisenberg effective Lagrangian supplemented with higher-order derivative corrections. We also examine the $\omega^2/m_e^2$ behavior predicted by including one-loop corrections to the Euler-Heisenberg effective Lagrangian together with the necessary counterterms. At the one-loop level, it is not possible to reproduce the $\omega^2/m_e^2$ dependence of QED by simply using the Euler-Heisenberg expansion