Reducible contributions to quantum electrodynamics in external fields

We consider one-particle reducible (1PR) contributions to QED and scalar QED processes in external fields, at one loop and two loops. We investigate three cases in detail: constant crossed fields, constant magnetic fields, and plane waves. We find that 1PR tadpole contributions in plane waves and constant crossed fields are non-zero, but contribute only divergences to be renormalised away. In constant magnetic fields, on the other hand, tadpole contributions give physical corrections to processes at one loop and beyond. Our calculations are exact in the external fields and we give weak field expansions in the magnetic case.