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

Definition

The vertex function ${\displaystyle \Gamma ^{\mu }}$ can be defined in terms of a functional derivative of the effective action S_eff as

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

The dominant (and classical) contribution to ${\displaystyle \Gamma ^{\mu }}$ is the gamma matrix ${\displaystyle \gamma ^{\mu }}$, 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:

${\displaystyle \Gamma ^{\mu }=\gamma ^{\mu }F_{1}(q^{2})+{\frac {i\sigma ^{\mu \nu }q_{\nu }}{2m}}F_{2}(q^{2})}$

where ${\displaystyle \sigma ^{\mu \nu }=(i/2)[\gamma ^{\mu },\gamma ^{\nu }]}$, ${\displaystyle q_{\nu }}$ is the incoming four-momentum of the external photon (on the right-hand side of the figure), and F₁(q²) and F₂(q²) are the Dirac and Pauli form factors, respectively, 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:

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

In 1948, Julian Schwinger calculated the first correction to anomalous magnetic moment, given by

${\displaystyle F_{2}(0)\approx {\frac {\alpha }{2\pi }}}$

where α is the fine-structure constant.

See also

• Nonoblique correction