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

TL;DR

This paper develops a formalism for calculating QED corrections to operator matrix elements involving heavy composite particles, with applications to nuclear and atomic physics, including beta decay and CKM matrix element extraction.

Contribution

It introduces a new classification of reducible and irreducible graphs and demonstrates their use in evaluating QED corrections to heavy particle systems.

Findings

Formalism for QED corrections to heavy particles established.

One-loop corrections expressed as operator matrix elements.

Application to superallowed beta decay and isospin breaking identified.

Abstract

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 $O(Z^2\alpha^2)$ 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.