Open AccessFeature PaperArticle
Theory of the Anomalous Magnetic Moment of the Electron
^{1}
Theory Center, IPNS, KEK, Tsukuba 3050801, Japan
^{2}
Nishina Center, RIKEN, Wako 3510198, Japan
^{3}
KobayashiMaskawa Institute for the Origin of Particles and the Universe, Nagoya University, Nagoya 4648602, Japan
^{4}
Laboratory for Elementary Particle Physics, Cornell University, Ithaca, NY 14853, USA
^{5}
Amherst Center for Fundamental Interactions, Department of Physics, University of Massachusetts, Amherst, MA 01003, USA
^{6}
Department of Physics, Saitama University, Saitama 3388570, Japan
^{*}
Author to whom correspondence should be addressed.
Received: 27 November 2018 / Revised: 3 February 2019 / Accepted: 12 February 2019 / Published: 22 February 2019

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Abstract
The anomalous magnetic moment of the electron
${a}_{e}$ measured in a Penning trap occupies a unique position among high precision measurements of physical constants in the sense that it can be compared directly with the theoretical calculation based on the renormalized quantum electrodynamics (QED) to high orders of perturbation expansion in the fine structure constant
$\alpha $ , with an effective parameter
$\alpha /\pi $ . Both numerical and analytic evaluations of
${a}_{e}$ up to
${(\alpha /\pi )}^{4}$ are firmly established. The coefficient of
${(\alpha /\pi )}^{5}$ has been obtained recently by an extensive numerical integration. The contributions of hadronic and weak interactions have also been estimated. The sum of all these terms leads to
${a}_{e}\left(\mathrm{theory}\right)$ =
$1\phantom{\rule{3.33333pt}{0ex}}159\phantom{\rule{3.33333pt}{0ex}}652\phantom{\rule{3.33333pt}{0ex}}181.606\phantom{\rule{3.33333pt}{0ex}}\left(11\right)\left(12\right)\left(229\right)\times {10}^{12}$ , where the first two uncertainties are from the tenthorder QED term and the hadronic term, respectively. The third and largest uncertainty comes from the current best value of the finestructure constant derived from the cesium recoil measurement:
${\alpha}^{1}\left(\mathrm{Cs}\right)=137.035\phantom{\rule{3.33333pt}{0ex}}999\phantom{\rule{3.33333pt}{0ex}}046\phantom{\rule{3.33333pt}{0ex}}\left(27\right)$ . The discrepancy between
${a}_{e}\left(\mathrm{theory}\right)$ and
${a}_{e}\left(\left(\mathrm{experiment}\right)\right)$ is 2.4
$\sigma $ . Assuming that the standard model is valid so that
${a}_{e}$ (theory) =
${a}_{e}$ (experiment) holds, we obtain
${\alpha}^{1}\left({a}_{e}\right)\phantom{\rule{3.33333pt}{0ex}}=137.035\phantom{\rule{3.33333pt}{0ex}}999\phantom{\rule{3.33333pt}{0ex}}1496\phantom{\rule{3.33333pt}{0ex}}\left(13\right)\left(14\right)\left(330\right)$ , which is nearly as accurate as
${\alpha}^{1}\left(\mathrm{Cs}\right)$ . The uncertainties are from the tenthorder QED term, hadronic term, and the best measurement of
${a}_{e}$ , in this order.
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