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 ψ, the antifermion , and the vector potential 'A'.

Contents

Definition

Reference

Definition

The vertex function $Gamma\_mu$ can be defined in terms of a functional derivative of the effective action Γ_eff as
:
(It is unfortunate that notationally, the effective action Γ_eff and the vertex function Γ^μ happen to share the same kernel symbol.)

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 transversity of the photon, as expressed by the Ward identity; and invariance under parity -- to take the following form:
:
u} q_{
u}}{2 m} F_2(q^2)
where $sigma^\{mu$
u} = (i/2) [gamma^{mu}, gamma^{
u}] , $q\_\{$
u} 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:
:

Reference

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