# 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 can be defined in terms of a functional derivative of the effective action S_{eff} as

The dominant (and classical) contribution to is the gamma matrix , 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 field strength renormalization. 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:

## Notes

## References

- Gross, F. (1993).
*Relativistic Quantum Mechanics and Field Theory*(1st ed.). Wiley-VCH. ISBN 978-0471591139. - Peskin, Michael E.; Schroeder, Daniel V. (1995).
*An Introduction to Quantum Field Theory*. Reading: Addison-Wesley. ISBN 0-201-50397-2. - Weinberg, S. (2002),
*Foundations*, The Quantum Theory of Fields,**I**, Cambridge University Press, ISBN 0-521-55001-7