Vertex function

Effective particle coupling beyond tree level

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

Definition

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

Γ μ = − 1 e δ 3 S e f f δ ψ ¯ δ ψ δ A μ {\displaystyle \Gamma ^{\mu }=-{1 \over e}{\delta ^{3}S_{\mathrm {eff} } \over \delta {\bar {\psi }}\delta \psi \delta A_{\mu }}}
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 Γ μ {\displaystyle \Gamma ^{\mu }} is the gamma matrix γ μ {\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:

Γ μ = γ μ F 1 ( q 2 ) + i σ μ ν q ν 2 m F 2 ( q 2 ) {\displaystyle \Gamma ^{\mu }=\gamma ^{\mu }F_{1}(q^{2})+{\frac {i\sigma ^{\mu \nu }q_{\nu }}{2m}}F_{2}(q^{2})}

where σ μ ν = ( i / 2 ) [ γ μ , γ ν ] {\displaystyle \sigma ^{\mu \nu }=(i/2)[\gamma ^{\mu },\gamma ^{\nu }]} , q ν {\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 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 ) {\displaystyle a={\frac {g-2}{2}}=F_{2}(0)}

References

  • Gross, F. (1993). Relativistic Quantum Mechanics and Field Theory (1st ed.). Wiley-VCH. ISBN 978-0471591139.
  • Peskin, Michael E.; Schroeder, Daniel V. (1995). An Introduction to Quantum Field Theory. Reading: Addison-Wesley. ISBN 0-201-50397-2.
  • Weinberg, S. (2002), Foundations, The Quantum Theory of Fields, vol. I, Cambridge University Press, ISBN 0-521-55001-7

External links

  • Media related to Vertex function at Wikimedia Commons


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