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 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 satisfies 0