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. 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 Seff as

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

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 transversality of the photon, as expressed by the Ward identity; and invariance under 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 {{Math|''F''1(''q''2)}} and {{Math|''F''2(''q''2)}} are the Dirac and Pauli form factors,<ref>{{Cite book |last=Wong |first=Samuel S. M. |url=https://www.google.fr/books/edition/Introductory_Nuclear_Physics/CrYyEQAAQBAJ?hl=en&gbpv=1&dq=dirac+pauli+form+factor&pg=PA113&printsec=frontcover |title=Introductory Nuclear Physics |date=2024-11-12 |publisher=John Wiley & Sons |isbn=978-3-527-41445-1 |language=en}}</ref> respectively, that depend only on the momentum transfer ''q''2. At tree level (or leading order), {{Math|''F''1(''q''2) {{=}} 1}} and {{Math|''F''2(''q''2) {{=}} 0}}. Beyond leading order, the corrections to {{Math|''F''1(0)}} are exactly canceled by the field strength renormalization. The form factor {{Math|''F''2(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> In 1948, Julian Schwinger calculated the first correction to anomalous magnetic moment, given by <blockquote><math> F_2(0)\approx \frac{\alpha}{2\pi} </math></blockquote>where ''α'' is the fine-structure constant.<ref>{{Cite journal |last=Teubner |first=Thomas |date=2018 |title=The anomalous anomaly |url=https://www.nature.com/articles/s41567-018-0341-3 |journal=Nature Physics |language=en |volume=14 |issue=11 |pages=1148–1148 |doi=10.1038/s41567-018-0341-3 |issn=1745-2481}}</ref>

==See also== *Nonoblique correction

==References== {{Reflist}} *{{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}}

Category:Quantum electrodynamics Category:Quantum field theory