# 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 ${\displaystyle \psi }$, the antifermion ${\displaystyle {\bar {\psi }}}$, and the vector potential A.

## Definition

The vertex function ${\displaystyle \Gamma ^{\mu }}$  can be defined in terms of a functional derivative of the effective action Seff as

${\displaystyle \Gamma ^{\mu }=-{1 \over e}{\delta ^{3}S_{\mathrm {eff} } \over \delta {\bar {\psi }}\delta \psi \delta A_{\mu }}}$

The one-loop correction to the vertex function. This is the dominant contribution to the anomalous magnetic moment of the electron.

The dominant (and classical) contribution to ${\displaystyle \Gamma ^{\mu }}$  is the gamma matrix ${\displaystyle \gamma ^{\mu }}$ , 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:

${\displaystyle \Gamma ^{\mu }=\gamma ^{\mu }F_{1}(q^{2})+{\frac {i\sigma ^{\mu \nu }q_{\nu }}{2m}}F_{2}(q^{2})}$

where ${\displaystyle \sigma ^{\mu \nu }=(i/2)[\gamma ^{\mu },\gamma ^{\nu }]}$ , ${\displaystyle q_{\nu }}$  is the incoming four-momentum of the external photon (on the right-hand side of the figure), and F1(q2) and F2(q2) are form factors that depend only on the momentum transfer q2. At tree level (or leading order), F1(q2) = 1 and F2(q2) = 0. Beyond leading order, the corrections to F1(0) are exactly canceled by the field strength renormalization. The form factor F2(0) corresponds to the anomalous magnetic moment a of the fermion, defined in terms of the Landé g-factor as:

${\displaystyle a={\frac {g-2}{2}}=F_{2}(0)}$

## 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