Recently, Gies and Karbstein showed that the two-loop Euler–Heisenberg Lagrangian receives a finite one-particle reducible contribution in addition to the well-known one-particle irreducible one. Here, we demonstrate that a similar contribution exists for the propagator in a constant field already at the one-loop level, and we calculate this contribution for the scalar QED case. We also present an independent derivation of the Gies–Karbstein result using the worldline formalism, treating the scalar and spinor QED cases in a unified manner.
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