# Vertex function

In quantum electrodynamics, the **vertex function** describes the coupling between a photon and an electron beyond the leading order of perturbation theory. In particular, it is the one particle irreducible correlation function involving the fermion , the antifermion , and the vector potential **A**.

## Definition

The vertex function Γ^{μ} can be defined in terms of a functional derivative of the effective action S_{eff} as

The dominant (and classical) contribution to Γ^{μ} is the gamma matrix γ^{μ}, which explains the choice of the letter. The vertex function is constrained by the symmetries of quantum electrodynamics — Lorentz invariance; gauge invariance or the transversality of the photon, as expressed by the Ward identity; and invariance under parity — to take the following form:

where , is the incoming four-momentum of the external photon (on the right-hand side of the figure), and F_{1}(q^{2}) and F_{2}(q^{2}) are *form factors* that depend only on the momentum transfer q^{2}. At tree level (or leading order), F_{1}(q^{2}) = 1 and F_{2}(q^{2}) = 0. Beyond leading order, the corrections to F_{1}(0) are exactly canceled by the wave function renormalization of the incoming and outgoing electron lines according to the Ward-Takahashi identity. The form factor F_{2}(0) corresponds to the anomalous magnetic moment *a* of the fermion, defined in terms of the Landé g-factor as:

## References

- Michael E. Peskin and Daniel V. Schroeder,
*An Introduction to Quantum Field Theory*, Addison-Wesley, Reading, 1995.