The anomalous magnetic moment of the muon in the Standard Model

Abstract: 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.

• The paper thoroughly analyzes QED, electroweak, and hadronic contributions to muon g-2, employing advanced computational methods to address precision challenges.
• The paper demonstrates a persistent 3–4 standard deviation discrepancy between Standard Model predictions and experimental measurements, hinting at new physics.
• The paper outlines future directions using lattice QCD and dispersive techniques to refine theoretical predictions and reduce uncertainties.

The Anomalous Magnetic Moment of the Muon in the Standard Model

The anomalous magnetic moment of the muon, denoted as $a_\mu \equiv (g-2)_\mu/2$, represents a critical parameter in particle physics, providing insights into the interactions described by the Standard Model (SM) and serving as a potential window into new physics beyond the current theoretical framework. The muon's $g-2$ is a fundamental property affected by electromagnetic interactions involving virtual particles, including those predicted but not directly observed.

Analyzing $a_\mu$ Within the Standard Model

The paper thoroughly reviews the state-of-the-art calculation of $a_\mu$ within the SM, encompassing contributions from Quantum Electrodynamics (QED), Electroweak theory, and Quantum Chromodynamics (QCD). Each of these sectors introduces different computational challenges and uncertainties:

1. QED Contributions: The first-order contribution in the perturbation expansion of the fine-structure constant, $\alpha$, is described by the Schwinger term, $\alpha/(2\pi)$. Higher-order QED corrections are computed with remarkable precision, limiting uncertainties primarily due to the yet uncalculated sixth-order terms.
2. Electroweak Contributions: These are suppressed by a factor of $(m_\mu/M_W)^2$, where $M_W$ is the mass of the W boson, and contribute smaller effects compared to QED. They have been evaluated to better than one percent precision up to two loops.
3. Hadronic Contributions: This sector introduces the most notable theoretical uncertainties and is dominated by hadronic vacuum polarization (HVP) effects at $\mathcal{O}(\alpha^2)$ and hadronic light-by-light (HLbL) scattering at $\mathcal{O}(\alpha^3)$. Dispersive methods and lattice QCD approaches are pivotal in estimating these contributions, with significant ongoing efforts to refine these calculations to reduce theoretical errors to match experimental progress.

The Current Status and Tension with Experiment

The tension between the experimental results, notably from the E821 experiment at Brookhaven National Laboratory, and theoretical predictions in the SM has persisted over the years, hovering around 3 to 4 standard deviations. Such discrepancies suggest potential contributions from physics beyond the SM, which could manifest through interactions involving heavy virtual particles not accounted for in the SM framework.

Experimental and Theoretical Advances

The experimental landscape is poised to refine the precision of $a_\mu$ further, with initiatives like the Fermilab Muon g-2 experiment (E989) aiming to decrease experimental uncertainties by a factor of four. Concurrently, the J-PARC experiment proposes an alternative approach using low-momentum muon beams to achieve comparable sensitivity.

The collaboration in the form of the Muon g-2 Theory Initiative underscores the community's concerted effort to consolidate the theoretical calculations with experimental analyses. The Initiative facilitates interactions across diverse groups working on hadronic contributions and their cross-verification against experimental data, a crucial aspect for reducing uncertainties in the SM prediction of $a_\mu$.

Conclusion and Future Prospects

The paper underlines the muon's anomalous magnetic moment as not merely a quantity of intrinsic interest but a sensitive probe for new physics. Future efforts in both theoretical computations and experimental measurements are vital. With improvements in lattice QCD calculations and experimental methodologies, together with sophisticated dispersive techniques, the endeavor continues to enhance the precision of $a_\mu$, potentially unlocking new realms of fundamental interactions. Such advances signify a critical phase in particle physics, where corroborating existing discrepancies with higher precision could ultimately redefine our understanding of the universe.