Vertex function
Template:Short description 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 <math>\psi</math>, the antifermion <math>\bar{\psi}</math>, and the vector potential A.
DefinitionEdit
The vertex function <math>\Gamma^\mu</math> can be defined in terms of a functional derivative of the effective action Seff as
- <math>\Gamma^\mu = -{1\over e}{\delta^3 S_{\mathrm{eff}}\over \delta \bar{\psi} \delta \psi \delta A_\mu}</math>
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 transversality of the photon, as expressed by the Ward identity; and invariance under 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 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 field strength renormalization. The form factor F2(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>