Editing Vertex function

{{short description|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 (quantum mechanics)|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'''.

==Definition==
The vertex function <math>\Gamma^\mu</math> can be defined in terms of a [[functional derivative]] of the [[effective action]] S<sub>eff</sub> as

:<math>\Gamma^\mu = -{1\over e}{\delta^3 S_{\mathrm{eff}}\over \delta \bar{\psi} \delta \psi \delta A_\mu}</math>

[[Image:vertex_correction.svg|thumb|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 <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 [[Photon polarization|transversality]] of the photon, as expressed by the [[Ward identity]]; and invariance under [[Parity (physics)|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 F<sub>1</sub>(q<sup>2</sup>) and F<sub>2</sub>(q<sup>2</sup>) are ''[[Form factor (quantum field theory)|form factor]]s'' that depend only on the momentum transfer q<sup>2</sup>.  At tree level (or leading order), F<sub>1</sub>(q<sup>2</sup>) = 1 and F<sub>2</sub>(q<sup>2</sup>) = 0.  Beyond leading order, the corrections to F<sub>1</sub>(0) are exactly canceled by the [[field strength renormalization]].  The form factor F<sub>2</sub>(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>

==See also==
*[[Nonoblique correction]]

==References==
*{{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}}

==External links==
*{{Commons category-inline}}

{{QED}}

[[Category:Quantum electrodynamics]]
[[Category:Quantum field theory]]


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