Vertex corrections and wavefunction renormalization for atoms, nuclei, and other heavy composite particles

We study QED corrections to operator matrix elements involving heavy composite particles (e.g., heavy-mesons, nuclei, and atoms). We define a new notion of reducible and irreducible graphs which is useful for systems with many discrete excited states. The equivalence of the LSZ reduction formula and old fashioned perturbation theory is explicitly demonstrated. The self energy and vertex corrections are defined (to all orders), and the one-loop corrections are reduced to operator matrix elements which may be evaluated by hadronic, nuclear, or atomic theorists. The gauge dependence of the various pieces are studied in detail at one loop, and cancellation of spurious contributions are demonstrated in a class of covariant gauges; Coulomb gauge is also discussed. The formalism is applied to superallowed beta decay where the one-loop structure is connected to existing literature based on current algebra techniques. We further identify the well known $O(Z^2α^2)$ isospin breaking correction from the intranuclear Coulomb field as arising from two-loop diagrams. We comment on future applications of our results to the radiative corrections necessary in extractions of $|V_{ud}|$, in particular for corrections that required beyond one-loop order.