We solve the Minkowski-space Schwinger-Dyson equation (SDE) for the fermion propagator in quantum electrodynamics (QED) with massive photons. Specifically, we work in the quenched approximation within the rainbow-ladder truncation. Loop-divergences are regularized by the Pauli-Villars regularization. With moderately strong fermion-photon coupling, we find that the analytic structure of the fermion propagator consists of an on-shell pole and branch-cuts located in the timelike region. Such structures are consistent with the direct solution of the fermion propagator as functions of the complex momentum. Our method paves the way towards the calculation of the Minkowski-space Bethe-Salpeter amplitude using dressed fermion propagator.
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