  It is known that the one-loop effective action of ${QED}_2$ is a quadratic in
the field strength when the fermion mass is zero: all potential higher order
contributions beyond second order vanish. For nonzero fermion mass it is shown
that this behavior persists for a general class of fields for at least one
value of the fermion mass when the external field's flux $\Phi$ satisfies
$0<|e\Phi|<2\pi$. For ${QED}_4$ the mass-shell renormalized one-loop effective
action vanishes for at least one value of the fermion mass for a class of
smooth, square integrable background gauge fields provided a plausible
zero-mass limit exists.
