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The Euclidean fermionic determinant in four-dimensional quantum electrodynamics is considered as a function of the fermionic mass for a class of $O(2)\times O(3)$ symmetric background gauge fields. These fields result in a determinant free of all cutoffs. Consider the one-loop effective action, the logarithm of the determinant, and subtract off the renormalization dependent second-order term. Suppose the small-mass behavior of this remainder is fully determined by the chiral anomaly. Then either the remainder vanishes at least once as the fermionic mass is varied in the interval $0<m<\infty$ or it reduces to its fourth-order value in which case the new remainder, obtained after subtracting the fourth-order term, vanishes at least once. Which possibility is chosen depends on the sign of simple integrals involving the field strength tensor and its dual.

Evidence for mass zeros of the fermionic determinant in four-dimensional quantum electrodynamics