  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.
