Vertex function

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

The vertex function Γμ can be defined in terms of a functional derivative of the effective action Γeff as

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

(It is unfortunate that the effective action Γeff and the vertex function Γμ happen to be described by the same letter.)

The one-loop correction to the vertex function. This is the dominant contribution to the anomalous magnetic moment of the electron.
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 Γμ is the gamma matrix γμ, which explains the choice of the letter. The vertex function is constrained by the symmetries of quantum electrodynamics -- Lorentz invariance; gauge invariance or the transversity 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 σμν = (i / 2)[γμ,γν], qν 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)

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