  Using the external field method, {\it i.e.\/} evaluating the effective action
$V_{\mathrm{eff}}$ for an arbitrarily strong constant and homogeneous field, we
explore nonperturbative properties of QED allowing arbitrary gyromagnetic ratio
$g$. We find a cusp at $g = 2$ in: a) The QED $b_0$-renormalization group
coefficient, and in the infinite wavelength limit in b) a subclass containing
the pseudoscalar ${\cal P}^{2n}= (\vec E\cdot\vec B)^{2n} $ of light-light
scattering coefficients. Properties of $b_0$ imply for certain domains of $g$
asymptotic freedom in an Abelian theory.
