  We explore nonperturbative properties of QED allowing a gyromagnetic ratio
$g\ne g_{\rm D}\equiv 2$. We study the effective action $V_{\mathrm{eff}}$ for
an arbitrarily strong constant and homogeneous field. Using the external field
method, we find a cusp as a function of the gyromagnetic factor $g$ in: a) The
QED $b_0$-renormalization group coefficient; b) A subclass of light-light
scattering coefficients obtained in the long wavelength limit expansion. We
recognize possibility of asymptotic freedom in an Abelian theory for certain
domains of $g$.
