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

Definition

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 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:

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

References

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

External links

Media related to Vertex function at Wikimedia Commons

v t e Quantum electrodynamics
Formalism	Euler–Heisenberg Lagrangian Feynman diagram Gupta–Bleuler formalism Path integral formulation
Particles	Dual photon Electron Faddeev–Popov ghost Photon Positron Positronium Virtual particles
Concepts	Anomalous magnetic dipole moment Furry's theorem Klein–Nishina formula Landau pole QED vacuum Self-energy Schwinger limit Uehling potential Vacuum polarization Vertex function Ward–Takahashi identity
Processes	Bhabha scattering Breit–Wheeler process Bremsstrahlung Compton scattering Delbrück scattering Lamb shift Møller scattering Schwinger effect Photon-photon scattering
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