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 Γ^μ can be defined in terms of a functional derivative of the effective action S_eff as

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

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 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:

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

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 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 Landé 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.