Vertex function explained

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

Definition

The vertex function

\Gamma ^{\mu} can be defined in terms of a functional derivative of the effective action S_eff as\Gamma ^{\mu} = - { 1 \over e}{\delta ^{3} S _{eff} \over \delta \bar{\psi} \delta \psi \delta A _{\mu}}

The dominant (and classical) contribution to

\Gamma ^{\mu} is the gamma matrix \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:\Gamma ^{\mu} = \gamma ^{\mu} F ^{2 )} _{1 (q} + (i \sigma ^{\mu\nu} q _{\nu})/(2 m) F ^{2 )} _{2 (q}

where

\sigma ^{\mu\nu} = (i/ 2 ) [\gamma ^{\mu} , \gamma ^{\nu} ] , 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 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:a = (g -2)/(2) = F _{2(0)}

See also

• Nonoblique correction

References

• Book: Gross, F.. Relativistic Quantum Mechanics and Field Theory. 1993. 1st. Wiley-VCH. 978-0471591139.
• Book: Peskin. Michael E.. Michael Peskin. Schroeder. Daniel V.. An Introduction to Quantum Field Theory. registration. Addison-Wesley. Reading. 1995. 0-201-50397-2.