Editing Vertex function (section)

==Definition==
The vertex function <math>\Gamma^\mu</math> can be defined in terms of a [[functional derivative]] of the [[effective action]] S<sub>eff</sub> as

:<math>\Gamma^\mu = -{1\over e}{\delta^3 S_{\mathrm{eff}}\over \delta \bar{\psi} \delta \psi \delta A_\mu}</math>

[[Image:vertex_correction.svg|thumb|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 <math>\Gamma^\mu</math> is the [[gamma matrix]] <math>\gamma^\mu</math>, which explains the choice of the letter.  The vertex function is constrained by the symmetries of quantum electrodynamics — [[Lorentz invariance]]; [[gauge invariance]] or the [[Photon polarization|transversality]] of the photon, as expressed by the [[Ward identity]]; and invariance under [[Parity (physics)|parity]] — to take the following form:

:<math> \Gamma^\mu = \gamma^\mu F_1(q^2) + \frac{i \sigma^{\mu\nu} q_{\nu}}{2 m} F_2(q^2) </math>

where <math> \sigma^{\mu\nu} = (i/2) [\gamma^{\mu}, \gamma^{\nu}] </math>, <math> q_{\nu} </math> is the incoming four-momentum of the external photon (on the right-hand side of the figure), and F<sub>1</sub>(q<sup>2</sup>) and F<sub>2</sub>(q<sup>2</sup>) are ''[[Form factor (quantum field theory)|form factor]]s'' that depend only on the momentum transfer q<sup>2</sup>.  At tree level (or leading order), F<sub>1</sub>(q<sup>2</sup>) = 1 and F<sub>2</sub>(q<sup>2</sup>) = 0.  Beyond leading order, the corrections to F<sub>1</sub>(0) are exactly canceled by the [[field strength renormalization]].  The form factor F<sub>2</sub>(0) corresponds to the [[anomalous magnetic moment]] ''a'' of the fermion, defined in terms of the [[Landé g-factor]] as:

:<math> a = \frac{g-2}{2} = F_2(0) </math>