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\section*{anomalous magnetic moment}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebraic_qunantum_field_theory}{}\paragraph*{{Algebraic Qunantum Field Theory}}\label{algebraic_qunantum_field_theory}

[[!include AQFT and operator algebra contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Anomalies}{Anomalies}\dotfill \pageref*{Anomalies} \linebreak
\noindent\hyperlink{contributions}{Contributions}\dotfill \pageref*{contributions} \linebreak
\noindent\hyperlink{qed_contributions}{QED contributions}\dotfill \pageref*{qed_contributions} \linebreak
\noindent\hyperlink{QuantumGravityCorrection}{Quantum gravity contributions}\dotfill \pageref*{QuantumGravityCorrection} \linebreak
\noindent\hyperlink{axion_contributions}{Axion contributions}\dotfill \pageref*{axion_contributions} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{experiment_and_deviation}{Experiment and deviation}\dotfill \pageref*{experiment_and_deviation} \linebreak
\noindent\hyperlink{qed_contributions_2}{QED contributions}\dotfill \pageref*{qed_contributions_2} \linebreak
\noindent\hyperlink{gravity_contributions}{Gravity contributions}\dotfill \pageref*{gravity_contributions} \linebreak
\noindent\hyperlink{ReferencesAxionContributions}{Axion contributions}\dotfill \pageref*{ReferencesAxionContributions} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[perturbative quantum field theory]], the [[magnetic moment]] of [[particles]] as predicted by [[classical field theory]] may receive corrections due to [[quantum physics|quantum effects]]. Such corrections are also called \emph{[[quantum anomalies]]}, and hence one speaks of the \emph{anomalous magnetic moments}, traditionally denoted by ``$g-2$''.

The archetypical example is the anomalous magnetic moment of the [[electron]] in [[quantum electrodynamics]], which is famous as the [[pQFT]]-prediction that matches [[experiment]] to an accuracy of about $10^{-12}$ (e.g.\hyperlink{Scharf95}{Scharf 95, (3.10.20)}).

Similarly there is the anomalous magenetic moment $g_\mu - 2$ of the [[muon]] and the other [[leptons]].

\hypertarget{Anomalies}{}\subsection*{{Anomalies}}\label{Anomalies}

In fact, the anomalous magnetic moment of the [[muon]] $g_\mu - 2$ has become notorious for apparently showing a noticeable \emph{discrepancy} between [[theory (physics)|theoretic prediction]] from the [[standard model of particle physics]] and its value as determined in [[experiment]]. The discrepancy is now found to have [[statistical significance]] around 3.5[[standard deviation|σ]] (\hyperlink{DHMZ17}{DHMZ 17}) or 4[[standard deviation|σ]] (\hyperlink{Jegerlehner18a}{Jegerlehner 18a}). Recent measurements may even show a possible deviation around 2.5 [[standard deviation|σ]] for the electron's anomalous magnetic moment (see \hyperlink{Falkowski18}{Falkowski 18}). Details depend on understanding of [[non-perturbative effects]] (\hyperlink{Jegerlehner18b}{Jegerlehner 18b, section 2}).

If these experimental ``anomalies'' (in the sense of [[phenomenology]]) in the anomalous magnetic moment $g_\mu - 2$ (and possibly even in $g_e -2$) are real (the established rule of thumb is that deviations are established once their [[statistical significance]] reaches 5[[standard deviation|σ]], see \href{statistical+significance#ParticlePhysics}{here}), they point to ``new physics'' beyond the [[standard model of particle physics]]. See also at \emph{[[flavour anomaly]]}.

In fact \hyperlink{Lyons13b}{Lyons 13b} argued that the detection-threshold of the [[statistical significance]] of anomalies here should be just 4[[standard deviation|σ]], which would mean that they should already count as being detected:

$\backslash$begin\{center\}  $\backslash$end\{center\}

\begin{quote}%
table taken from \hyperlink{Lyons13b}{Lyons 13b, p. 4}


\end{quote}
Possible explanations for the anomomalies in the anomalous magnetic moments is the existence of [[leptoquarks]] (\hyperlink{BauerNeubert15}{Bauer-Neubert 15}, \hyperlink{CCDM16}{CCDM 16}, \hyperlink{Falkowski17}{Falkowski 17}, \hyperlink{Mueller18}{Müller 18}), which at the same time are a candidate for explaining the [[flavour anomalies]] (see also \hyperlink{ChiangOkada17}{Chiang-Okada 17}).

\hypertarget{contributions}{}\subsection*{{Contributions}}\label{contributions}

\hypertarget{qed_contributions}{}\subsubsection*{{QED contributions}}\label{qed_contributions}

(\ldots{}) for the [[electron]] see e.g. \hyperlink{Scharf95}{Scharf 95, section 3.10} (\ldots{})

\hypertarget{QuantumGravityCorrection}{}\subsubsection*{{Quantum gravity contributions}}\label{QuantumGravityCorrection}

The further corrections of [[loop order|1-loop]] [[perturbative quantum gravity]] to the anomalous magnetic moment of the [[electron]] and the [[muon]] have been computed in (\hyperlink{BerendsGastman75}{Berends-Gastman 75}) and found to be finite without need for [[renormalization]]. These [[Feynman diagrams]] contribute:



\hypertarget{axion_contributions}{}\subsubsection*{{Axion contributions}}\label{axion_contributions}

Possible contributions to and xconstraints on $g_{lep}-2$ from hypothetical [[axions]] are discussed in \hyperlink{ACGM08}{ACGM 08}, \hyperlink{MMPP16}{MMPP 16}, \hyperlink{BNT17}{BNT 17}\ldots{}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[electric dipole moment]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Basic discussion:

\begin{itemize}%
\item [[Othmar Steinmann]], \emph{What is the Magnetic Moment of the Electron?}, Commun.Math.Phys. 237 (2003) 181-201 (\href{https://arxiv.org/abs/hep-ph/0211187}{arXiv:hep-ph/0211187})


\item Kirill Melnikov, Arkady Vainshtein, \emph{Theory of the Muon Anomalous Magnetic Moment}, Springer Tracts in Modern Physics 216, 2006


\item [[Friedrich Jegerlehner]], \emph{The Anomalous Magnetic Moment of the Muon}, Springer Tracts in Modern Physics 226, Springer-Verlag Berlin Heidelberg, 2008



\end{itemize}
Discussion of detection-threshold for the [[statistical significance]] of anomalies:

\begin{itemize}%
\item [[Louis Lyons]], \emph{Discovering the Significance of 5 sigma} (\href{https://arxiv.org/abs/1310.1284}{arXiv:1310.1284})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment}{Anomalous magnetic dipole moment}}

\end{itemize}
\hypertarget{experiment_and_deviation}{}\subsubsection*{{Experiment and deviation}}\label{experiment_and_deviation}

Discussion of precision experiment and possible deviation from theory:

\begin{itemize}%
\item Michel Davier, Andreas Hoecker, Bogdan Malaescu, Zhiqing Zhang, \emph{Reevaluation of the hadronic vacuum polarisation contributions to the Standard Model predictions of the muon g-2 and alpha(mZ) using newest hadronic cross-section data}, Eur. Phys. J. C (2017) 77: 827 (\href{https://arxiv.org/abs/1706.09436}{arXiv:1706.09436})


\item J. L. Holzbauer on behalf of the Muon g-2 collaboration, \emph{The Muon g-2 Experiment Overview and Status}, Proceedings for The 19th International Workshop on Neutrinos from Accelerators (NUFACT 2017) (\href{https://arxiv.org/abs/1712.05980}{arXiv:1712.05980})


\item [[Fred Jegerlehner]], \emph{The Muon g-2 in Progress}, Acta Physica Polonica 2018 (\href{https://arxiv.org/ct?url=https%3A%2F%2Fdx.doi.org%2F10.5506%252FAPhysPolB.49.1157&v=01073571}{doi:10.5506/APhysPolB.49.1157}, \href{https://arxiv.org/abs/1804.07409}{arXiv:1804.07409})


\item [[Fred Jegerlehner]], \emph{The Role of Mesons in Muon $g-2$} (\href{https://arxiv.org/abs/1809.07413}{arXiv:1809.07413})


\item [[Adam Falkowski]], \emph{\href{http://resonaances.blogspot.com/2018/06/alpha-and-g-minus-two.html}{Both $g-2$ anomalies}}, June 2018



\end{itemize}
Possible explanation of the anomaly in the anomalous magnetic moments in terms of [[leptoquarks]]:

\begin{itemize}%
\item Martin Bauer, Matthias Neubert, \emph{One Leptoquark to Rule Them All: A Minimal Explanation for $R_{D^{(\ast)}}$, $R_K$ and $(g-2)_\mu$}, Phys. Rev. Lett. 116, 141802 (2016) (\href{https://arxiv.org/abs/1511.01900}{arXiv:1511.01900})


\item Estefania Coluccio Leskow, Andreas Crivellin, Giancarlo D'Ambrosio, Dario Müller, \emph{$(g-2)_\mu$, Lepton Flavour Violation and Z Decays with Leptoquarks: Correlations and Future Prospects}, Phys. Rev. D 95, 055018 (2017) (\href{https://arxiv.org/abs/1612.06858}{arXiv:1612.06858})


\item \{BiswasShaw19\} Anirban Biswas, Avirup Shaw, \emph{Reconciling dark matter, $R_{K^{(\ast)}}$ anomalies and $(g-2)_\mu$ in an $L_\mu-L_\tau$ scenario} (\href{https://arxiv.org/abs/1903.08745}{arXiv:1903.08745})


\item [[Adam Falkowski]], \emph{\href{http://resonaances.blogspot.com/2015/11/leptoquarks-strike-back.html}{Leptoquarks strike back}}, November 2017


\item Cheng-Wei Chiang, Hiroshi Okada, \emph{A simple model for explaining muon-related anomalies and dark matter} (\href{https://arxiv.org/abs/1711.07365}{arXiv:1711.07365})


\item Dario Müller, \emph{Leptoquarks in Flavour Physics}, EPJ Web of Conferences 179, 01015 (2018) (\href{https://arxiv.org/abs/1801.03380}{arXiv:1801.03380})


\item Junichiro Kawamura, Stuart Raby, Andreas Trautner, \emph{Complete Vector-like Fourth Family and new $U(1)'$ for Muon Anomalies} (\href{https://arxiv.org/abs/1906.11297}{arXiv:1906.11297})



\end{itemize}
\hypertarget{qed_contributions_2}{}\subsubsection*{{QED contributions}}\label{qed_contributions_2}

The computation of the anomalous magnetic dipole moment of the [[electron]] in [[QED]] is spelled out (via [[causal perturbation theory]]) in

\begin{itemize}%
\item [[Günter Scharf]], section 3.10, culminating in (3.10.20), of \emph{[[Finite Quantum Electrodynamics -- The Causal Approach]]}, Berlin: Springer-Verlag, 1995, 2nd edition

\end{itemize}
\hypertarget{gravity_contributions}{}\subsubsection*{{Gravity contributions}}\label{gravity_contributions}

Corrections at [[loop order|1-loop]] from [[quantum gravity]] are discussed in

\begin{itemize}%
\item F. A. Berends, R. Gastmans, \emph{Quantum gravity and the electron and muon anomalous magnetic moment}, Phys. Lett. B55 Issue 3 Feb 1975 311-312 ()

\end{itemize}
This discussion is adapted to [[supergravity]] in

\begin{itemize}%
\item F. del Aguila, A. Culatti, R. Munoz-Tapia, M. Perez-Victoria, \emph{Supergravity corrections to $(g-2)_l$ in differential renormalization}, Nuclear Physics B 504 (1997) 532-550 (\href{https://arxiv.org/abs/hep-ph/9702342}{arXiv:hep-ph/9702342})

\end{itemize}
\hypertarget{ReferencesAxionContributions}{}\subsubsection*{{Axion contributions}}\label{ReferencesAxionContributions}

Contribution of hypothetical [[axions]] to the [[anomalous magnetic moment]] of the [[electron]] and [[muon]] in [[QED]]:

\begin{itemize}%
\item Yannis Semertzidis, \emph{Magnetic and Electric Dipole Moments in Storage Rings}, chapter 6 of Markus Kuster, Georg Raffelt, Berta Beltrán (eds.), \emph{Axions: Theory, cosmology, and Experimental Searches}, Lect. Notes Phys. 741 (Springer, Berlin Heidelberg 2008) ()


\item Roberta Armillis, Claudio Coriano, Marco Guzzi, Simone Morelli, \emph{Axions and Anomaly-Mediated Interactions: The Green-Schwarz and Wess-Zumino Vertices at Higher Orders and g-2 of the muon}, JHEP 0810:034,2008 (\href{https://arxiv.org/abs/0808.1882}{arXiv:0808.1882})


\item W.J. Marciano, A. Masiero, P. Paradisi, M. Passera, \emph{Contributions of axion-like particles to lepton dipole moments}, Phys. Rev. D 94, 115033 (2016) (\href{https://arxiv.org/abs/1607.01022}{arXiv:1607.01022})


\item Martin Bauer, Matthias Neubert, Andrea Thamm, \emph{Collider Probes of Axion-Like Particles}, J. High Energ. Phys. (2017) 2017: 44. (\href{https://arxiv.org/abs/1708.00443}{arXiv:1708.00443}, )



\end{itemize}
The basic relevant [[Feynman diagrams]] are worked out here:

\begin{itemize}%
\item \href{http://www-personal.umich.edu/~jbourj/peskin/6-3.pdf}{pdf}

\end{itemize}
[[!redirects anomalous magnetic moments]]

[[!redirects anomalous magnetic dipole moment]] [[!redirects anomalous magnetic dipole moments]]

[[!redirects electron anomalous magnetic moment]] [[!redirects muon anomalous magnetic moment]]

[[!redirects anomalous magnetic moment of the electron]] [[!redirects anomalous magnetic dipole moment of the electron]]

[[!redirects g-2]]



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