We review the present status of the Standard Model calculation of the anomalous magnetic moment of the muon. This is performed in a perturbative expansion in the fine-structure constant $\alpha$ and is broken down into pure QED, electroweak, and hadronic contributions. The pure QED contribution is by far the largest and has been evaluated up to and including $\mathcal{O}(\alpha^5)$ with negligible numerical uncertainty. The electroweak contribution is suppressed by $(m_\mu/M_W)^2$ and only shows up at the level of the seventh significant digit. It has been evaluated up to two loops and is known to better than one percent. Hadronic contributions are the most difficult to calculate and are responsible for almost all of the theoretical uncertainty. The leading hadronic contribution appears at $\mathcal{O}(\alpha^2)$ and is due to hadronic vacuum polarization, whereas at $\mathcal{O}(\alpha^3)$ the hadronic light-by-light scattering contribution appears. Given the low characteristic scale of this observable, these contributions have to be calculated with nonperturbative methods, in particular, dispersion relations and the lattice approach to QCD. The largest part of this review is dedicated to a detailed account of recent efforts to improve the calculation of these two contributions with either a data-driven, dispersive approach, or a first-principle, lattice-QCD approach. The final result reads $a_\mu^\text{SM}=116\,591\,810(43)\times 10^{-11}$ and is smaller than the Brookhaven measurement by 3.7$\sigma$. The experimental uncertainty will soon be reduced by up to a factor four by the new experiment currently running at Fermilab, and also by the future J-PARC experiment. This and the prospects to further reduce the theoretical uncertainty in the near future-which are also discussed here-make this quantity one of the most promising places to look for evidence of new physics.

中文翻译：

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标准模型中μ子的异常磁矩

我们回顾了μ子异常磁矩的标准模型计算的现状。这是在精细结构常数 $\alpha$ 的微扰扩展中执行的，并被分解为纯 QED、电弱和强子贡献。纯 QED 贡献是迄今为止最大的，并且已经被评估到并包括 $\mathcal{O}(\alpha^5)$，数值不确定性可以忽略不计。弱电贡献被 $(m_\mu/M_W)^2$ 抑制，仅显示在第七位有效数字的水平。它已被评估了多达两个循环，并且已知优于百分之一。强子贡献是最难计算的，几乎所有的理论不确定性都是由它造成的。主要的强子贡献出现在 $\mathcal{O}(\alpha^2)$ 并且是由于强子真空极化，而在 $\mathcal{O}(\alpha^3)$ 是强子逐光散射贡献出现。鉴于此可观察量的低特征尺度，必须使用非微扰方法计算这些贡献，特别是色散关系和 QCD 的晶格方法。本综述的最大部分致力于详细说明最近通过数据驱动的色散方法或第一原理的格子 QCD 方法改进这两个贡献的计算的努力。最终结果为 $a_\mu^\text{SM}=116\,591\,810(43)\times 10^{-11}$，比 Brookhaven 测量值小 3.7$\sigma$。费米实验室目前正在运行的新实验以及未来的 J-PARC 实验将很快将实验不确定性降低四倍。这一点以及在不久的将来进一步减少理论不确定性的前景——这里也讨论了——使这个数量成为寻找新物理学证据的最有希望的地方之一。