Muon Magnetic Anomaly

• Muon Magnetic Anomaly is the deviation of the muon’s gyromagnetic ratio from 2, serving as a precise test of quantum electrodynamics and the Standard Model.
• Experimental techniques measure this anomaly via muon spin precession in uniform magnetic fields using high-precision NMR and calorimetry to achieve sub-ppm accuracy.
• The observed tension between experimental results and theoretical predictions drives investigations into new physics models, such as supersymmetry and dark sector candidates.

The muon magnetic anomaly, denoted $a_\mu = (g_\mu-2)/2$, is a fundamental experimental and theoretical quantity in particle physics that quantifies the deviation of the muon's gyromagnetic ratio, $g_\mu$, from the Dirac value of 2. Precise measurement and calculation of $a_\mu$ provide stringent tests of the Standard Model (SM), constrain radiative corrections, and serve as sensitive probes of physics beyond the SM. In recent years, measurements at Fermilab (E989) and Brookhaven (E821) have demonstrated a persistent tension between the experimental value of $a_\mu$ and its SM prediction, reaching a significance of over $4\sigma$ with current global averages. This anomaly is dominated by uncertainties in QCD-driven hadronic vacuum polarization, with new experimental approaches and theoretical developments aiming to resolve the discrepancy.

1. Formal Definition and Theoretical Decomposition

The anomalous magnetic moment is defined as: $a_\mu = \frac{g_\mu - 2}{2}$ for a spin-½ particle of mass $m_\mu$ and charge $q$: $\vec{\mu}_\mu = g_\mu \frac{q}{2m_\mu} \vec{S}$

Within the Standard Model, the anomaly is decomposed as: $$a_\mu^{\rm SM} = a_\mu^{\rm QED} + a_\mu^{\rm EW} + a_\mu^{\rm HVP} + a_\mu^{\rm HLbL}$$ where:

• $a_\mu^{\rm QED}$: Quantum electrodynamics contributions, computed up to 5 loops, providing the dominant term ($>99\%$), e.g. $$a_\mu^{\rm QED} = 116584718.931(0.104) \times 10^{-11}$$ (Malaescu, 2022).
• $a_\mu^{\rm EW}$: Electroweak loops (W, Z, Higgs), small but precisely determined ($\simeq 153.6(1.0) \times 10^{-11}$) (Malaescu, 2022).
• $a_\mu^{\rm HVP}$: Hadronic vacuum polarization, a leading source of theory uncertainty ($\simeq 6845(40) \times 10^{-11}$).
• $a_\mu^{\rm HLbL}$: Hadronic light-by-light scattering, a subleading but non-negligible contribution ($\simeq 92(18) \times 10^{-11}$).

Total SM value (Theory Initiative, 2020): $a_\mu^{\rm SM} = 116\,591\,810\,(43)\times10^{-11}$ with error dominated by the hadronic contributions (Pocanic, 1 Dec 2025, Malaescu, 2022).

2. Experimental Techniques and Key Observables

The leading method for measuring $a_\mu$ exploits its imprint on the spin precession of muons stored in a uniform magnetic field $B$. The anomalous precession frequency: $$\omega_a = \omega_s - \omega_c = a_\mu \frac{qB}{m_\mu}$$ is extracted from the time- and energy-dependent distribution of decay positrons via: $$N(t) = N_0 e^{-t/\tau_\mu} [1 + A \cos(\omega_a t + \phi)]$$ where higher-energy positrons, preferentially emitted along the muon-spin direction, are detected by an array of segmented electromagnetic calorimeters.

The magnetic field is measured by nuclear magnetic resonance (NMR) probes: a precision in-vacuum trolley carrying 17 probes performs spatial mapping every few days, while 378 fixed probes monitor field drift. The average field seen by muons is expressed via the equivalent proton Larmor frequency $\omega_p$, calibrated by a water-based reference probe (Tewsley-Booth, 2022, Albahri et al., 2021).

The final determination utilizes the ratio: $$a_\mu = \frac{\omega_a}{\langle \omega_p \rangle_\mu} \cdot \frac{\mu_p}{\mu_\mu} - 1$$ with all quantities referenced to fundamental constants and controlled for systematics at the sub-ppm level.

3. Systematics, Corrections, and Error Control

Achieving high precision in $a_\mu$ requires rigorous control of systematic uncertainties:

Source	Typical magnitude (Run-1/2/3)	Key controls
Magnetic field calibration	56–70 ppb	Trolley/fixed NMR cross-calibrate
Transient fields (ESQ, kickers)	37–92 ppb	Dedicated probes, mapping models
Muon distribution weighting	11–40 ppb	Straw trackers, profile modeling
Electric field & pitch effects	40 ppb	Momentum selection, quad scans
Calorimeter gain/timing	25–30 ppb	Laser calibration, segmentation
Pileup correction	25 ppb	Pulse-fit algorithms
Lost muon effects	15 ppb	Collimator scans, fit models

Total systematics from all sources are controlled to $\simeq100$ ppb in current data, with further hardware upgrades and analysis refinements targeting 140 ppb for the experiment’s final precision (Sorbara, 12 Mar 2025, Albahri et al., 2021).

Beam-dynamics corrections (electric field, pitch angle, muon loss, phase acceptance) are explicitly quantified via dedicated simulation and data-driven studies, yielding additive corrections at the few-hundred-ppb level (Lucà, 2022).

4. Experimental Results and Statistical Tension

The global average of E821 (BNL) and E989 (Fermilab) data gives: $$a_\mu^{\rm exp} = 116\,592\,061\,(41) \times 10^{-11} \quad (0.35~\text{ppm})$$ compared to the SM value $a_\mu^{\rm SM}$ above, yielding: $$\Delta a_\mu = a_\mu^{\rm exp} - a_\mu^{\rm SM} = (251 \pm 59) \times 10^{-11}$$ corresponding to a $4.2\sigma$ excess over the SM (Pocanic, 1 Dec 2025, Keshavarzi et al., 2021, Tewsley-Booth, 2022). Current uncertainties are balanced between statistical and systematic, with each at $\simeq 29 \times 10^{-11}$ (Pocanic, 1 Dec 2025).

Recent lattice QCD determinations (BMW 2024) of the HVP contribution are consistent with experiment within $1\sigma$, potentially reconciling the anomaly. Data-driven dispersive approaches (based on $e^+e^- \to$ hadrons cross-sections) maintain the $4–5\sigma$ discrepancy (Pocanic, 1 Dec 2025).

5. Theoretical Interpretation and New Physics Scenarios

The observed tension in $\Delta a_\mu$ strongly motivates models of beyond-the-Standard-Model physics that generate loop-level corrections. Principal candidates include:

• Supersymmetry (SUSY): Smuon–neutralino and sneutrino–chargino loops can naturally generate $\Delta a_\mu$ of the observed size for superpartner masses $\sim 100–800$ GeV, and large $\tan\beta$ ($40–60$) (Babu et al., 2021, Ahmed et al., 2021, Zhang et al., 2021). Collider searches at HL-LHC and HE-LHC can probe the relevant parameter space.
• Dark photons and new gauge bosons: Sub-GeV U(1) bosons with millicharge couplings yield loop contributions of the correct order of magnitude (Malaescu, 2022).
• Leptoquarks, two-Higgs-doublet models, flavor-violating sectors: These offer viable loop shifts but are increasingly constrained by LHC and flavor-observables (Keshavarzi et al., 2021).

Stringent bounds arise from global fits to LHC, flavor physics, and dark-matter relic density. Multi-loop extensions (e.g., Barr-Zee diagrams) and Higgs-sneutrino mixing further provide testable collider signatures (e.g., altered $h\to\mu\mu$ branching ratios) (Zhang et al., 2021).

6. Future Directions: MUonE and Complementary Experiments

With ongoing Run-4 to Run-6 analysis at Fermilab expected to push experimental precision below 0.2 ppm, definitive answers are anticipated in the near term. The CERN MUonE experiment aims to determine the leading-order HVP contribution via muonic Bhabha scattering, expected to reach $<0.5\%$ precision in the next several years and provide an independent check on dispersive/lattice inputs (Pocanic, 1 Dec 2025).

Parallel programs:

• J-PARC E34 (Japan): alternative running-momentum muon ring (Keshavarzi et al., 2021)
• Precision flavor, EDM, and dark-sector searches: MEG II, Mu3e, Mu2e, NA64μ, M³ (Keshavarzi et al., 2021)
• High-statistics $e^+e^-\to$ hadrons data from CMD-3, BaBar, KLOE, BESIII, Belle II

The combination of experimental and theoretical advances places the muon anomaly at the crossroads of SM verification and the search for new physics.

7. Summary Table: Key Numbers and Uncertainties

Parameter	Value ($\times 10^{-11}$)	Uncertainty ($\times 10^{-11}$)
$a_\mu^{\rm exp}$ (BNL+FNAL)	116592061	41
$a_\mu^{\rm SM}$ (Theory Initiative, 2020)	116591810	43
$\Delta a_\mu$	251	59
QED (5-loop)	116584718.931	0.104
EW	153.6	1.0
HVP (data-driven, TIWP)	6845	40
HLbL	92	18
Experimental goal (Fermilab Run-6, projected)	—	<20

Current status: With lattice-QCD-based HVP, the $\Delta a_\mu$ anomaly is potentially resolved, but data-driven approaches and low-energy $e^+e^-\to$ hadrons data still support an enduring tension. Continued improvements in both experimental measurement and theoretical calculation of hadronic contributions are critical for a decisive resolution of the muon magnetic anomaly.