Zeros of the Fredholm determinant for the external-field problem in Euclidean, massless spin-0 and spin-1/2 electrodynamics

I study the location of zeros, in the plane of complex coupling constant $e$, of the Fredholm determinant for the external-field problem in Euclidean spin-0 and spin-1/2 electrodynamics. In the massive spin-0 and spin-1/2 cases, a simple argument gives the already known result that the Fredholm determinant has no zeros for real $\mathrm{eA}$. This result extends to the massless spin-0 case. However, I construct a class of counterexamples which shows that there are zeros for real $\mathrm{eA}$, with $A$ not a pure gauge potential, in massless spin-1/2 electrodynamics. The functional measure of the potentials $A$ obtained from the construction is not determined. The construction is based on the fact that the problem of determining zero eigenvalues in an external field in massless spin-1/2 electrodynamics separates into two uncoupled two-component equations. I discuss possible implications of this result for the study of spinor electrodynamics by nonperturbative path-integral methods.