vertex function

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

:$\backslash Gamma^\backslash mu\; =\; -\{1\backslash over\; e\}\{\backslash delta^3\; S\_\{\backslash mathrm\{eff\}\}\backslash over\; \backslash delta\; \backslash bar\{\backslash psi\}\; \backslash delta\; \backslash psi\; \backslash delta\; A\_\backslash mu\}$

Image:vertex_correction.svg

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

:$\backslash Gamma^\backslash mu\; =\; \backslash gamma^\backslash mu\; F\_1(q^2)\; +\; \backslash frac\{i\; \backslash sigma^\{\backslash mu\backslash nu\}\; q\_\{\backslash nu\}\}\{2\; m\}\; F\_2(q^2)$

where $\backslash sigma^\{\backslash mu\backslash nu\}\; =\; (i/2)\; [\backslash gamma^\{\backslash mu\},\; \backslash gamma^\{\backslash nu\}]$, $q\_\{\backslash nu\}$ 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:

:$a\; =\; \backslash frac\{g-2\}\{2\}\; =\; F\_2(0)$

• Nonoblique correction

• {{cite book|last=Gross|first=F.|title=Relativistic Quantum Mechanics and Field Theory|year=1993|edition=1st|publisher=Wiley-VCH|isbn=978-0471591139}}
• {{cite book|last1=Peskin|first1=Michael E.|authorlink1=Michael Peskin|last2=Schroeder|first2=Daniel V.|title=An Introduction to Quantum Field Theory|url=https://archive.org/details/introductiontoqu0000pesk|url-access=registration|publisher=Addison-Wesley|location=Reading|year=1995|isbn=0-201-50397-2}}
• {{citation|last=Weinberg|first=S.|authorlink=Steven Weinberg|year=2002|title=Foundations|series=The Quantum Theory of Fields|volume=I|isbn=0-521-55001-7|publisher=Cambridge University Press|url-access=registration|url=https://archive.org/details/quantumtheoryoff00stev}}

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Category:Quantum electrodynamics

Category:Quantum field theory

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