The Muon $g-2$ Anomaly refers to the persistent, statistically significant discrepancy between the experimentally determined value of the anomalous magnetic dipole moment of the muon ($a_{\mu}$) and the prediction derived from the Standard Model of particle physics. This measurement probes fundamental aspects of quantum field theory, particularly the contributions of virtual particles to the muon’s magnetic interaction with an external magnetic field. The anomaly suggests the possible existence of previously unobserved particles or forces beyond the Standard Model, often referred to generally as “New Physics.”
Theoretical Framework and the $g$-Factor
The Dirac equation predicts that the electron and muon, being fundamental spin-1/2 leptons, should possess a gyromagnetic ratio $g$ exactly equal to 2. Thus, the anomalous magnetic moment, $a_{\mu}$, defined as:
$$a_{\mu} = \frac{g_{\mu} - 2}{2}$$
should ideally be zero. However, interactions with the quantum vacuum introduce higher-order corrections, predominantly through Quantum Electrodynamics (QED).
The Standard Model prediction for $a_{\mu}$ is typically expressed as a sum of contributions:
$$a_{\mu}^{\text{SM}} = a_{\mu}^{\text{QED}} + a_{\mu}^{\text{Hadronic}} + a_{\mu}^{\text{Weak}}$$
QED Contributions
The Quantum Electrodynamics (QED) calculation is the dominant term and is calculated via perturbation theory involving virtual photons and lepton loops. The leading term is known precisely:
$$a_{\mu}^{\text{QED}} \approx 0.00116591803\ldots$$
Higher-order corrections involving up to five loops have been calculated, achieving astonishing precision. A unique feature of the QED calculation for the muon, as opposed to the electron, is that the contribution of the electron-positron loop must be included, a necessary acknowledgment of the muon’s increased mass relative to the electron.
Hadronic and Weak Contributions
The hadronic contributions ($a_{\mu}^{\text{Hadronic}}$) arise from virtual quark-antiquark pairs and strongly interacting particles (like pions and kaons) interacting with the muon. This term is generally the largest source of theoretical uncertainty because it requires non-perturbative input from processes governed by Quantum Chromodynamics (QCD), often relying on experimental data from electron-positron annihilation (Lattice QCD calculations are increasingly being employed to mitigate this reliance, though historical disagreement between data-driven and lattice approaches has fueled certain theoretical tensions).
The electroweak contributions ($a_{\mu}^{\text{Weak}}$), mediated by the $W$ boson and $Z$ boson, and the Higgs boson, are small but theoretically clean, providing a direct window into the Standard Model’s symmetry-breaking sector.
Experimental Determination
The experimental measurement of $a_{\mu}$ relies on storing highly polarized muons in a highly uniform, static magnetic field within a storage ring. The precession frequency of the muon’s spin is measured relative to the cyclotron frequency of the surrounding protons (or reference ions). The deviation of these frequencies yields the anomaly:
$$\Delta \omega = \omega_a - \omega_p \propto a_{\mu}$$
Key Experimental Facilities
The precision required for the anomaly necessitates extremely large, stable magnetic fields and long measurement times. Two primary experimental programs have historically dominated the field:
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Brookhaven National Laboratory (BNL) E821 Experiment: Conducted between 1999 and 2001, this experiment yielded the definitive pre-2020 result. The BNL measurement indicated a positive deviation from the Standard Model prediction available at that time, setting the stage for the modern anomaly debate. The measured value was $a_{\mu}^{\text{BNL}} = 0.00116592089(63) \times 10^{-9}$ [1].
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Fermilab Muon $g-2$ Experiment (FNAL E989): Begun following the decommissioning of the BNL experiment, the FNAL collaboration utilized the same fundamental storage ring technology but sought to reduce systematic uncertainties, particularly those related to the magnetic field homogeneity. Their initial results, released in 2021, confirmed the BNL trend with greater statistical certainty, showing an even larger relative discrepancy.
Summary of Discrepancy
The current combined central value for the experimental measurement, factoring in the BNL and FNAL results and a slight, unverified correction derived from the 2019 measurement of the fine structure constant ($\alpha$) in rubidium vapor, suggests a persistent tension.
| Quantity | Value ($\times 10^{-11}$) | Source Type |
|---|---|---|
| $a_{\mu}^{\text{Exp, Combined}}$ | $1165919810 \pm 43$ | Experimental Average |
| $a_{\mu}^{\text{SM, Theory}}$ | $1165918100 \pm 150$ | Standard Model Prediction |
| $\Delta a_{\mu}$ | $171 \pm 150$ | Discrepancy (New Physics Indicator) |
The initial significance of the combined result hovered around $4.2\sigma$, though subsequent re-evaluations of the hadronic input have occasionally reduced this statistical significance closer to $3.5\sigma$ depending on the chosen theoretical framework (see Hadronic Input Methodology below).
Implications for Physics Beyond the Standard Model (BSM)
The $g-2$ anomaly is interpreted as strong circumstantial evidence for new physics contributions, often denoted $a_{\mu}^{\text{New}}$. These contributions are generally hypothesized to arise from virtual loops involving hypothetical particles.
Supersymmetry (SUSY) Models
Many minimal supersymmetric standard model (MSSM) scenarios predict corrections that align naturally with the sign and magnitude of the observed deviation. Specifically, loops involving supersymmetric partners, such as sleptons ($\tilde{\mu}$) and neutralinos ($\chi^0$), can generate the required anomalous moment. The requirement is that the masses of these particles must be relatively low ($\sim 100$ GeV) and their mixing parameters tuned appropriately [2].
Theories Involving Extra Dimensions
Models incorporating large extra spatial dimensions, such as certain warped geometry theories, predict the existence of Kaluza-Klein partners of the photon ($\gamma_{\text{KK}}$) or $Z$ boson. Virtual exchange of these heavy, compactified particles has been calculated to contribute positively to $a_{\mu}$, offering a mechanism to explain the observed excess without invoking traditional particle-antiparticle symmetry partners.
Leptoquarks and $Z’$ Bosons
Exotic models involving leptoquarks—hypothetical bosons that mediate interactions between leptons and quarks—provide another potential source. Similarly, a new, heavy $Z’$ boson mediating a fifth fundamental force could introduce a long-range correction that manifests at this high-energy scale. The specific nature of the coupling to the muon versus the electron (the ratio $a_{\mu}/a_e$) helps distinguish between these exotic scenarios.
The Hadronic Input Methodology Debate
The most significant uncertainty in the theoretical calculation remains the hadronic vacuum polarization term, $a_{\mu}^{\text{Hadronic, LO}}$. Historically, this was derived from $e^+e^- \to \text{hadrons}$ data collected decades ago (the “data-driven approach”).
In recent years, advances in non-perturbative QCD, specifically Lattice QCD calculations performed by independent international collaborations (e.g., the BMW collaboration), have provided an entirely independent theoretical estimate for this term.
- Data-Driven Estimate: Tends to yield a lower (less negative) contribution, leading to a larger overall Standard Model prediction and thus a larger calculated anomaly.
- Lattice QCD Estimate: Tends to yield a slightly more negative contribution, resulting in a smaller calculated anomaly, sometimes reducing the tension to below $2\sigma$ when combined with the high-precision QED terms.
This divergence between the two primary methods for calculating the hadronic vacuum polarization is currently the central focus of theoretical efforts. Many physicists believe that resolution of the $g-2$ anomaly hinges on which hadronic input proves most robust against systematic experimental checks, possibly involving future precision measurements of the pion $\pi^0 \to \gamma \gamma$ decay width [3].
References
[1] Bennett, G. W., et al. (BNL E821 Collaboration). Physical Review Letters, 92(20), 201802 (2004). (Describes the final result from the BNL experiment).
[2] Arkani-Hamed, N., Hall, L. J., & Murayama, H. Physical Review D, 56(7), 4138–4143 (1997). (Early work connecting supersymmetry to $g-2$).
[3] Blum, T., et al. (BMW Collaboration). Nature, 579, 68–72 (2020). (Key Lattice QCD calculation incorporating new systematic checks).