Vertex function

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

[edit] Definition

The vertex function $Γ μ$ can be defined in terms of a functional derivative of the effective action Γ_eff as

$\Gamma^\mu = -{1\over e}{\delta^3 \Gamma_{eff}\over \delta \bar{\psi} \delta \psi \delta A_\mu}$

(It is unfortunate that notationally, the effective action Γ_eff and the vertex function Γ^μ happen to share the same kernel symbol.)

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:

$\Gamma^\mu = \gamma^\mu F_1(q^2) + \frac{i \sigma^{\mu\nu} q_{\nu}}{2 m} F_2(q^2)$

where $σ μν = (i / 2)[γ μ,γ ν]$ , $q ν$ 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 Lande g-factor as:

$a = \frac{g-2}{2} = F_2(0)$

[edit] References

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

v • d • e Quantum Electrodynamics

electron • positron • photon • self-energy • vacuum polarization • vertex function • Gupta-Bleuler formalism • ξ gauge • Ward-Takahashi identity • Compton scattering • Bhabha scattering • Møller scattering • anomalous magnetic dipole moment • bremsstrahlung • positronium