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 Γ^μ 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 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 is the gamma matrix γ^μ, 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 wave function renormalization of the incoming and outgoing electron lines according to the Ward–Takahashi identity. 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

Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, Reading, 1995.

Quantum electrodynamics

Anomalous magnetic dipole moment Bhabha scattering Bremsstrahlung Compton scattering Electron Gupta–Bleuler formalism Møller scattering Photon Positron Positronium QED vacuum Self-energy Vacuum polarization Virtual particles Vertex function Ward–Takahashi identity ξ gauge

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