Mass zeros in the one-loop effective actions of QED in 1+1 and 3+1 dimensions

It is known that the one-loop effective action of QED₂ is a quadratic in the field strength when the fermion mass is zero: all potential higher order contributions beyond second order vanish. For a 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 Φ satisfies 0<|eΦ|<2π. For QED₄ with (anti-)self-dual fields the mass-shell renormalized one-loop effective action vanishes for at least one value of the fermion mass provided a plausible zero-mass limit exists.