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**.

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 F_{1}(q^{2}) and F_{2}(q^{2}) are *form factors* that depend only on the momentum transfer q^{2}. At tree level (or leading order), F_{1}(q^{2}) = 1 and F_{2}(q^{2}) = 0. Beyond leading order, the corrections to F_{1}(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 F_{2}(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
)