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

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


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

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


 * $$ \Gamma^\mu = \gamma^\mu F_1(q^2) + \frac{i \sigma^{\mu\nu} q_{\nu}}{2 m} F_2(q^2) $$

where $$ \sigma^{\mu\nu} = (i/2) [\gamma^{\mu}, \gamma^{\nu}] $$, $$ q_{\nu} $$ 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:


 * $$ a = \frac{g-2}{2} = F_2(0) $$