Effective Lagrangians and low energy photon-photon scattering

We use the behavior of the photon-photon scattering for photon energies $\omega$ less than the electron mass, $m_e,$ to examine the implications of treating the Euler-Heisenberg Lagrangian as an effective field theory. Specifically, we determine the ${\omega }^2{/m}_e^2$ behavior of the scattering amplitude predicted by including oneloop corrections to the Euler-Heisenberg effective Lagrangian together with the counterterms required by renormalizability. This behavior is compared with the energy dependence obtained by expanding the exact QED photon-photon scattering amplitude. If the introduction of counterterms in the effective field theory is restricted to those determined by renormalizability, the ${\omega }^2{/m}_e^2$ dependences of the two expansions differ.