Vertex function - Alchetron, The Free Social Encyclopedia

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 ψ , the antifermion ψ ¯ , 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

Γ μ = − 1 e δ 3 S e f f δ ψ ¯ δ ψ δ A μ

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:

Γ μ = γ μ F 1 ( q 2 ) + i σ μ ν q ν 2 m F 2 ( q 2 )

where σ μ ν = ( i / 2 ) [ γ μ , γ ν ] , q ν 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 = g − 2 2 = F 2 ( 0 )

References

Vertex function Wikipedia

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