  We address the issue of large-order expansions in strong-field QED. Our
approach is based on the one-loop effective action encoded in the associated
photon polarisation tensor. We concentrate on the simple case of crossed fields
aiming at possible applications of high-power lasers to measure vacuum
birefringence. A simple next-to-leading order derivative expansion reveals that
the indices of refraction increase with frequency. This signals normal
dispersion in the small-frequency regime where the derivative expansion makes
sense. To gain information beyond that regime we determine the factorial growth
of the derivative expansion coefficients evaluating the first 80 orders by
means of computer algebra. From this we can infer a nonperturbative imaginary
part for the indices of refraction indicating absorption (pair production) as
soon as energy and intensity become (super)critical. These results compare
favourably with an analytic evaluation of the polarisation tensor asymptotics.
Kramers-Kronig relations finally allow for a nonperturbative definition of the
real parts as well and show that absorption goes hand in hand with anomalous
dispersion for sufficiently large frequencies and fields.
