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Vertex function

Effective particle coupling beyond tree level

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Source updated August 21, 2026

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. But the use of a vertex function is not limited to QED.

01Definition

The vertex function $\Gamma ^{\mu }$ 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 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:

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

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 the Dirac and Pauli form factors, respectively, 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:

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

In 1948, Julian Schwinger calculated the first correction to anomalous magnetic moment, given by

$F_{2}(0)\approx {\frac {\alpha }{2\pi }}$

where α is the fine-structure constant.

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This article is adapted from the Wikipedia article “Vertex function”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Vertex function

01Definition

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