  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 one-loop corrections to the Euler-Heisenberg
effective Lagrangian together with the counterterms required by
renomalizability. 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.
