Inquantum electrodynamics, thevertex function describes the coupling between aphoton and anelectron beyond the leading order ofperturbation theory. In particular, it is theone particle irreducible correlation function involving thefermion${\displaystyle \psi }$, the antifermion${\displaystyle \overline{\psi }}$, and thevector potentialA.

The vertex function${\displaystyle {\mathrm{\Gamma }}^{\mu }}$ can be defined in terms of afunctional derivative of theeffective action S_eff as

${\displaystyle {\mathrm{\Gamma }}^{\mu }=-\frac{1}{e}\frac{{\delta }^3{S}_{\mathrm{e}\mathrm{f}\mathrm{f}}}{\delta \overline{\psi }\delta \psi \delta A_{\mu }}}$

The dominant (and classical) contribution to${\displaystyle {\mathrm{\Gamma }}^{\mu }}$ is thegamma matrix${\displaystyle {\gamma }^{\mu }}$, which explains the choice of the letter. The vertex function is constrained by the symmetries of quantum electrodynamics —Lorentz invariance;gauge invariance or thetransversality of the photon, as expressed by theWard identity; and invariance underparity — to take the following form:

${\displaystyle {\mathrm{\Gamma }}^{\mu }={\gamma }^{\mu }{F}_1(q^2)+\frac{i{\sigma }^{\mu \nu }{q}_{\nu }}{2m}{F}_2(q^2)}$

where${\displaystyle {\sigma }^{\mu \nu }=(i/2)[{\gamma }^{\mu },{\gamma }^{\nu }]}$,${\displaystyle q_{\nu }}$ is the incoming four-momentum of the external photon (on the right-hand side of the figure), andF₁(q²) andF₂(q²) are the Dirac and Pauliform factors,^[1] respectively, that depend only on the momentum transferq². At tree level (or leading order),F₁(q²) = 1 andF₂(q²) = 0. Beyond leading order, the corrections toF₁(0) are exactly canceled by thefield strength renormalization. The form factorF₂(0) corresponds to theanomalous magnetic momenta of the fermion, defined in terms of theLandé g-factor as:

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

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

${\displaystyle F_2(0)\approx \frac{\alpha }{2\pi }}$

whereα is thefine-structure constant.^[2]

• Nonoblique correction

• Gross, F. (1993).Relativistic Quantum Mechanics and Field Theory (1st ed.).Wiley-VCH.ISBN 978-0471591139.
• Peskin, Michael E.; Schroeder, Daniel V. (1995).An Introduction to Quantum Field Theory. Reading: Addison-Wesley.ISBN 0-201-50397-2.
• Weinberg, S. (2002),Foundations, The Quantum Theory of Fields, vol. I,Cambridge University Press,ISBN 0-521-55001-7

• Media related toVertex function at Wikimedia Commons

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