# Vertex function

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Effective particle coupling beyond tree level

In [quantum electrodynamics](/source/Quantum_electrodynamics), the **vertex function** describes the coupling between a [photon](/source/Photon) and an [electron](/source/Electron) beyond the leading order of [perturbation theory](/source/Perturbation_theory_(quantum_mechanics)). In particular, it is the [one particle irreducible correlation function](/source/One_particle_irreducible_correlation_function) involving the [fermion](/source/Fermion) ψ {\displaystyle \psi } , the antifermion ψ ¯ {\displaystyle {\bar {\psi }}} , and the [vector potential](/source/Vector_potential) **A**.

## Definition

The vertex function Γ μ {\displaystyle \Gamma ^{\mu }} can be defined in terms of a [functional derivative](/source/Functional_derivative) of the [effective action](/source/Effective_action) Seff as

- Γ μ = − 1 e δ 3 S e f f δ ψ ¯ δ ψ δ A μ {\displaystyle \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 {\displaystyle \Gamma ^{\mu }} is the [gamma matrix](/source/Gamma_matrix) γ μ {\displaystyle \gamma ^{\mu }} , which explains the choice of the letter. The vertex function is constrained by the symmetries of quantum electrodynamics — [Lorentz invariance](/source/Lorentz_invariance); [gauge invariance](/source/Gauge_invariance) or the [transversality](/source/Photon_polarization) of the photon, as expressed by the [Ward identity](/source/Ward_identity); and invariance under [parity](/source/Parity_(physics)) — to take the following form:

- Γ μ = γ μ F 1 ( q 2 ) + i σ μ ν q ν 2 m F 2 ( q 2 ) {\displaystyle \Gamma ^{\mu }=\gamma ^{\mu }F_{1}(q^{2})+{\frac {i\sigma ^{\mu \nu }q_{\nu }}{2m}}F_{2}(q^{2})}

where σ μ ν = ( i / 2 ) [ γ μ , γ ν ] {\displaystyle \sigma ^{\mu \nu }=(i/2)[\gamma ^{\mu },\gamma ^{\nu }]} , q ν {\displaystyle q_{\nu }} is the incoming four-momentum of the external photon (on the right-hand side of the figure), and *F*1(*q*2) and *F*2(*q*2) are the Dirac and Pauli [form factors](/source/Form_factor_(quantum_field_theory)),[1] respectively, that depend only on the momentum transfer *q*2. At tree level (or leading order), *F*1(*q*2) = 1 and *F*2(*q*2) = 0. Beyond leading order, the corrections to *F*1(0) are exactly canceled by the [field strength renormalization](https://en.wikipedia.org/w/index.php?title=Field_strength_renormalization&action=edit&redlink=1). The form factor *F*2(0) corresponds to the [anomalous magnetic moment](/source/Anomalous_magnetic_moment) *a* of the fermion, defined in terms of the [Landé g-factor](/source/Land%C3%A9_g-factor) as:

- a = g − 2 2 = F 2 ( 0 ) {\displaystyle a={\frac {g-2}{2}}=F_{2}(0)}

In 1948, [Julian Schwinger](/source/Julian_Schwinger) calculated the first correction to anomalous magnetic moment, given by

F 2 ( 0 ) ≈ α 2 π {\displaystyle F_{2}(0)\approx {\frac {\alpha }{2\pi }}}

where *α* is the [fine-structure constant](/source/Fine-structure_constant).[2]

## See also

- [Nonoblique correction](/source/Nonoblique_correction)

## References

1. **[^](#cite_ref-1)** Wong, Samuel S. M. (2024-11-12). [*Introductory Nuclear Physics*](https://www.google.fr/books/edition/Introductory_Nuclear_Physics/CrYyEQAAQBAJ?hl=en&gbpv=1&dq=dirac+pauli+form+factor&pg=PA113&printsec=frontcover). John Wiley & Sons. [ISBN](/source/ISBN_(identifier)) [978-3-527-41445-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-527-41445-1).

1. **[^](#cite_ref-2)** Teubner, Thomas (2018). ["The anomalous anomaly"](https://www.nature.com/articles/s41567-018-0341-3). *Nature Physics*. **14** (11): 1148–1148. [doi](/source/Doi_(identifier)):[10.1038/s41567-018-0341-3](https://doi.org/10.1038%2Fs41567-018-0341-3). [ISSN](/source/ISSN_(identifier)) [1745-2481](https://search.worldcat.org/issn/1745-2481).

- Gross, F. (1993). *Relativistic Quantum Mechanics and Field Theory* (1st ed.). [Wiley-VCH](/source/Wiley-VCH). [ISBN](/source/ISBN_(identifier)) [978-0471591139](https://en.wikipedia.org/wiki/Special:BookSources/978-0471591139).

- [Peskin, Michael E.](/source/Michael_Peskin); Schroeder, Daniel V. (1995). [*An Introduction to Quantum Field Theory*](https://archive.org/details/introductiontoqu0000pesk). Reading: Addison-Wesley. [ISBN](/source/ISBN_(identifier)) [0-201-50397-2](https://en.wikipedia.org/wiki/Special:BookSources/0-201-50397-2).

- [Weinberg, S.](/source/Steven_Weinberg) (2002), [*Foundations*](https://archive.org/details/quantumtheoryoff00stev), The Quantum Theory of Fields, vol. I, [Cambridge University Press](/source/Cambridge_University_Press), [ISBN](/source/ISBN_(identifier)) [0-521-55001-7](https://en.wikipedia.org/wiki/Special:BookSources/0-521-55001-7)

## External links

- Media related to [Vertex function](https://commons.wikimedia.org/wiki/Category:Vertex_function) at Wikimedia Commons

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