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| 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'''.
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| ==Definition==
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| The vertex function Γ<sup>μ</sup> can be defined in terms of a [[functional derivative]] of the [[effective action]] S<sub>eff</sub> as
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| :<math>\Gamma^\mu = -{1\over e}{\delta^3 S_{\mathrm{eff}}\over \delta \bar{\psi} \delta \psi \delta A_\mu}</math> | |
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| [[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.]]
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| The dominant (and classical) contribution to Γ<sup>μ</sup> is the [[gamma matrix]] γ<sup>μ</sup>, 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:
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| :<math> \Gamma^\mu = \gamma^\mu F_1(q^2) + \frac{i \sigma^{\mu\nu} q_{\nu}}{2 m} F_2(q^2) </math> | |
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| 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 factors'' 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 [[wave function renormalization]] of the incoming and outgoing electron lines according to the [[Ward-Takahashi identity]]. 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:
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| :<math> a = \frac{g-2}{2} = F_2(0) </math>
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| ==References==
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| *Michael E. Peskin and Daniel V. Schroeder, ''An Introduction to Quantum Field Theory'', Addison-Wesley, Reading, 1995.
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| {{QED}}
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| [[Category:Quantum electrodynamics]]
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| [[Category:Quantum field theory]]
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| {{quantum-stub}}
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Nice to meet you, I am Marvella Shryock. Managing people is his profession. California is our beginning location. What I love doing is taking part in baseball but I haven't produced a dime with it.
my blog post :: www.Asianas.com
