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 ψ, 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 Γ_eff as

:

(It is unfortunate that notationally, the effective action Γ_eff and the vertex function Γ^μ happen to share the same kernel symbol.)

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:

:$Gamma^mu\; =\; gamma^mu\; F\_1(q^2)\; +\; frac\{i\; sigma^\{mu\; u\}\; q\_\{\; u\{2\; m\}\; F\_2(q^2)$

where $sigma^\{mu\; u\}\; =\; (i/2)\; [gamma^\{mu\},\; gamma^\{\; u\}]$, $q\_\{\; u\}$ 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 wave function renormalization of the incoming and outgoing electron lines according to the Ward-Takahashi identity. The form factor F₂(0) corresponds to the anomalous magnetic moment "a" of the fermion, defined in terms of the Lande g-factor as:

:$a\; =\; frac\{g-2\}\{2\}\; =\; F\_2(0)$

References

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