Vertex function

 ( Quantum Electrodynamics ) 

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

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

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

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 $\backslash sigma^\{\backslash mu\backslash nu\}\; =\; (i/2)\; \backslash gamma^\{\backslash mu\},$, $q\_\{\backslash nu\}$ is the incoming four-momentum of the external photon (on the right-hand side of the figure), and and are the Dirac and Pauli form factors,

respectively, that depend only on the momentum transfer q². At tree level (or leading order), and . Beyond leading order, the corrections to are exactly canceled by the field strength renormalization. The form factor corresponds to the anomalous magnetic moment a of the fermion, defined in terms of the Landé g-factor as:

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

In 1948, Julian Schwinger calculated the first correction to anomalous magnetic moment, given by

$F\_2(0)\backslash approx\; \backslash frac\{\backslash alpha\}\{2\backslash pi\}$

where α is the fine-structure constant.

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See also

• Nonoblique correction

• F. Gross (1993). 9780471591139, Wiley-VCH. ISBN 9780471591139
• (1995). 9780201503975, Addison-Wesley. . ISBN 9780201503975

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External links

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