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

The vertex function ${\displaystyle \Gamma ^{\mu }}${\displaystyle \Gamma ^{\mu }} can be defined in terms of a functional derivative of the effective action S_eff as

${\displaystyle \Gamma ^{\mu }=-{1 \over e}{\delta ^{3}S_{\mathrm {eff} } \over \delta {\bar {\psi }}\delta \psi \delta A_{\mu }}}${\displaystyle \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 ${\displaystyle \Gamma ^{\mu }}${\displaystyle \Gamma ^{\mu }} is the gamma matrix ${\displaystyle \gamma ^{\mu }}${\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 the transversality of the photon, as expressed by the Ward identity; and invariance under parity — to take the following form:

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

where ${\displaystyle \sigma ^{\mu \nu }=(i/2)[\gamma ^{\mu },\gamma ^{\nu }]}${\displaystyle \sigma ^{\mu \nu }=(i/2)[\gamma ^{\mu },\gamma ^{\nu }]}, ${\displaystyle q_{\nu }}${\displaystyle 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 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:

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

• 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

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