Massless electrodynamics in the one-photon-mode approximation

We discuss single-fermion-loop vacuum-polarization processes in massless quantum electrodynamics in the one-photon-mode approximation, in which the fermion self-interacts (to all orders in perturbation theory) by the exchange of virtual photons in a single virtual-photon eigenmode. The isolation of one photon mode is made possible by using the O(5)-covariant formulation of massless QED introduced in two earlier papers, in which the photon wave operator has a discrete, rather than a continuous, spectrum. The amplitude integral formalism introduced previously expresses the one-mode radiative-corrected vacuum polarization in terms of the uncorrected vacuum amplitude in the presence of a one-mode external field. By exploiting the residual SO(3) \ifmmode\times\else\texttimes\fi{} O(2) symmetry of the one-mode external-field problem, which permits separation of variables, we reduce the external-field problem to a set of two coupled ordinary first-order differential equations. We show that when the two independent solutions to these equations are suitably standardized, their Wronskian gives (up to a constant factor) the external-field-problem Fredholm determinant. We study the distribution of zeros and asymptotic behavior of the Fredholm determinant, relate these properties to the coupling-constant analyticity of the one-mode vacuum polarization, and conclude by giving a brief list of unresolved questions.