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 $\bar$, 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

:$\Gamma^\mu = -$

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:

:$\Gamma^\mu = \gamma^\mu F_1(q^2) + \frac F_2(q^2)$

where \sigma^ = (i/2) gamma^, \gamma^, $q_$ 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:

:$a = \frac = F_2(0)$

References

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External links

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Quantum electrodynamics
Quantum field theory

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