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The analytic properties of the vertex functions for massless scalar and massless and massive spinor electrodynamics are investigated. First, a general tensor form for the vertex consistent with the Ward identities and free of kinematic singularities is constructed. All of the scalar functions that appear are then calculated to one-loop order in perturbation theory. It is found that in both massive and massless theories the infrared singularities appear in the longitudinal parts of the vertex. The massless theories have additional photon mass-shell singularities which appear in the transverse terms. Complete expressions for each vertex are obtained which contain only one nonelementary function.
Ball et al. (Sat,) studied this question.