# Muon g−2

Laplace’s Mini World · Plate F38 · Energy, Flight & Sensing · Sensing

> Muons circle a 15 m ring, their spins turning a little faster than their paths. Counting the positrons they leave measures that little, to 127 parts per billion.

- Interactive model: https://world.lapla.si/en/muong2/
- Embed: https://world.lapla.si/embed/en/muong2/
- 中文: https://world.lapla.si/zh/muong2/
- Updated: 2026-10-22

## 01 127 Parts per Billion

On 3 June 2025 Fermilab’s Muon g−2 experiment announced its final result: the muon’s magnetic anomaly is a\_μ = 0.001 165 920 705, with a relative uncertainty of 127 ppb, 127 parts per billion, better than the 140 ppb the experiment set out to reach.

The data come from six runs between 2018 and 2023, about 300 billion positrons analysed. The statistical uncertainty is 98 ppb and the systematic 78 ppb. Against Brookhaven’s 0.54 ppm of 2006, the world average is now more than four times as precise.

*a\_μ = 116 592 0705(148) × 10⁻¹² · 127 ppb (Fermilab, 2025)*

## 02 g Is Not 2

A muon, like an electron, has charge and spin, and so is a tiny magnet. How strong a magnet for its spin is written as the g-factor. Dirac’s equation for the electron, in 1928, gave g = 2. The muon is about 207 times heavier than the electron and otherwise much like it.

But virtual particles keep appearing and vanishing around any particle, and they push g a little above 2. The extra is written a = (g − 2)/2, the magnetic anomaly. In 1948 Julian Schwinger worked out its first term, α/2π ≈ 0.001 161. The known electromagnetic, weak and strong interactions all add a share; unknown heavy particles, if they exist, would in most models affect the muon about 43,000 times more than the electron.

*g = 2(1 + a) · a\_μ ≈ 0.001 165 92 · first term α/2π ≈ 0.001 161*

## 03 The Spin Runs Ahead

In a vertical, uniform magnetic field a muon runs round a circle, its momentum turning once per lap. Its spin turns too. If g were exactly 2, the two would turn together; with g a little above 2, the spin turns a little more each lap and gets further and further ahead.

The rate at which it gains, ω\_a = a·eB/m, is the anomalous precession frequency. It depends only on the anomaly a and the field B, not on the muon’s momentum. In this ring a lap takes about 149 ns, the spin gains about 12.3° per lap, a whole turn in about 29.3 laps, a period of 4.365 µs.

*ω\_a = ω\_s − ω\_c = a·eB/m · f\_a ≈ 229.08 kHz · 12.3° gained per lap*

## 04 The Magic Momentum

With a magnetic field alone, muons slowly spread out up and down. Electrostatic quadrupole plates push them back: four stations, each a 13° and a 26° section, 43 % of the circumference, at about 18 kV. But an electric field turns the spin as well, adding a term to ω\_a with the factor a − 1/(γ² − 1).

At γ = 29.3, a momentum of 3.094 GeV/c, that term vanishes: the electric field holds the muons and leaves the precession alone. This is the magic momentum. At 1.45 T a muon with it runs on a circle of radius 7.112 m, and that is what set the ring’s size. CERN’s third g−2 experiment, in the 1970s, was the first to use it.

*a − 1/(γ² − 1) = 0 ⇒ γ = 29.3, p = 3.094 GeV/c · R = p/(eB) = 7.112 m*

## 05 Where the Positrons Go

A μ⁺ decays into a positron and two neutrinos, after 2.197 µs on average at rest; in the ring it moves so close to the speed of light that its time runs 29.3 times slower, 64.4 µs on average, about 430 laps. The neutrinos fly off almost untouched by matter: see the plate on neutrinos.

The decay breaks parity: seen by the muon, the positron comes out more often along the spin. In the lab, positrons thrown forward along the momentum have the most energy. So when spin and momentum point the same way, there are most high-energy positrons. Lighter and with less energy than the muon, the positrons bend more sharply in the field, curl inwards and land in the 24 calorimeters inside the ring.

*μ⁺ → e⁺ ν\_e ν̄\_μ · 2.197 µs at rest, 64.4 µs in the ring*

## 06 The Wiggle

Each injection, a fill, lasts about 700 µs; 16 fills come every 1.4 s, and in 2018 each stored about 5,000 muons. The calorimeters record every positron’s time and energy. Count only those above about 1.7 GeV, lay them out in time, and the number falls with the 64.4 µs lifetime, with a rise and fall every 4.365 µs on top.

The wiggle is the spin swinging past the momentum: N(t) = N₀ e^(−t/γτ) \[1 + A cos(ω\_a t + φ)\], with A about 0.37. Fitting that to the counts gives ω\_a. The curve at the top right is simulated and fitted live by this page’s toy model.

*N(t) = N₀ e^(−t/γτ) \[1 + A cos(ω\_a t + φ)\] · γτ = 64.4 µs · 2π/ω\_a = 4.365 µs*

## 07 A Ratio of Two Frequencies

ω\_a gives only a·B, so the field must be known as precisely. It too is measured as a frequency: the precession of protons in the field, ω\_p, by nuclear magnetic resonance, about 61.79 MHz at 1.45 T.

a\_μ = (ω\_a / ω̃′\_p) · (μ′\_p / μ\_B) · (m\_μ / m\_e). The last two ratios come from other precise measurements. During the analysis the digitisers’ clock was deliberately set off by a secret amount, revealed only when the analysis was finished. For precision measurement by comparing frequencies, see also the plate on optical lattice clocks.

*a\_μ = (ω\_a / ω̃′\_p) · (μ′\_p / μ\_B) · (m\_μ / m\_e) · ω\_p/2π ≈ 61.79 MHz*

## 08 Cut Open: the C Magnet

The magnet’s cross-section is a C, 1.57 m high and 1.39 m wide, open towards the ring’s centre so that positrons curling inwards meet no iron on their way. Upper and lower yokes are joined by spacer plates at the back; in the opening, a row of poles above and below, 18 cm apart.

Three cryostats hold the superconducting coils: the outer one two windings, above and below, of 24 turns each; an inner one above and one below. The conductor is niobium–titanium in copper, clad in pure aluminium, at about 5.0 K and some 5,200 A. Between the poles sits the aluminium vacuum chamber, with the quadrupole plates and the 9 cm storage region.

*1.57 m high · 1.39 m wide · gap 18 cm · storage region 9 cm across*

## 09 Even to Parts per Million

The yoke comes in 12 sectors of 30°, about 57 t each, some 680 t in all. The 72 poles, 36 above and 36 below, are 56 cm wide, made to 50 µm in width and 40 µm in thickness; the gap between them varies by ±23 µm rms.

After shimming, the field over the storage region is even to about 14 ppm rms. The upper and lower pole surfaces each also carry 100 concentric correction coils, bringing the variation across the aperture, averaged round the ring, below 1 ppm. The yokes are wrapped in an insulating blanket, and they and the electronics are held to ±0.5 °C. Superconducting magnets also feature on the plate on ATLAS and the Higgs boson.

*About 680 t of iron · 72 poles · about 14 ppm rms after shimming*

## 10 The Trolley

Inside the vacuum chamber runs a trolley with 17 nuclear magnetic resonance probes at its front. Two cables pull it along rails once round the ring, measuring the field at about 9,000 places, every 3 to 7 days.

In between, about 400 fixed probes above and below the chamber keep track of the field’s drift. The probes hold petroleum jelly; their readings are calibrated against a standard probe of ultrapure water, in terms of the precession of protons in water.

*17 probes · about 9,000 places · every 3–7 days · about 400 fixed probes*

## 11 Injection, Then a Kick

The muons come from decaying pions: along a 279 m beamline about 80 % of the pions decay into muons, which then pass through a 505 m Delivery Ring. That decay breaks parity too, so nearly all the muons’ spins point the same way. A bunch about 120 ns long enters through a hole in the yoke’s back and passes the 1.7 m superconducting inflector, whose field cancels the main field so the muons enter almost in a straight line.

They leave it 77 mm outside the ideal orbit. A quarter turn on, three kicker magnets, 1.27 m each, fire together and kick the muons outwards by about 10.8 mrad, onto the storage orbit; they must be off again within the 149 ns the muons take to come round.

*77 mm off · kicked 10.8 mrad at 90° · off within 149 ns*

## 12 Calorimeters and Trackers

Each of the 24 calorimeters is an array of lead fluoride crystals, 9 columns by 6 rows, each 2.5 × 2.5 × 14 cm, 1,296 in all. A positron showers in the crystals and makes Cherenkov light, read by silicon photomultipliers sampled about 800 million times a second through the whole fill. A laser system keeps watch on every crystal’s gain.

At 180° and 270°, straw trackers stand in front of a calorimeter: 8 modules at each, every module 128 straws of aluminised Mylar, 5 mm across. Tracing the positrons’ tracks back to where they started shows how the muon beam is spread in the storage region, and how that changes with time.

*24 × 54 = 1,296 lead fluoride crystals · two tracker stations of 8 × 128 straws*

## 13 From CERN to Fermilab

CERN began measuring the muon’s g−2 in 1959. Its third experiment took data in 1974–1976 with a 14 m storage ring and the magic momentum, and published in 1979 at 7.3 ppm. Brookhaven’s E821 built this superconducting ring and published its final result in 2006, at 0.54 ppm.

In 2013 the ring’s superconducting coils and their cryostats, 17 t (60 t with the frame), went to Fermilab in one piece: they set off on 22 June by truck and barge, down the US east coast, round Florida, across the Gulf of Mexico and up inland waterways including the Tennessee–Tombigbee, and rolled into Fermilab at 4:07 a.m. on 26 July, 3,200 miles in 35 days. The iron yoke was taken apart and trucked separately. The experiment started up on 31 May 2017.

*1979 CERN 7.3 ppm · 2006 Brookhaven 0.54 ppm · 2013 the move · 2025 Fermilab 127 ppb*

## 14 The Theory Side

In 2020 the Muon g−2 Theory Initiative gave a Standard Model prediction of 116 591 810(43) × 10⁻¹¹, its hardest part, from the strong interaction, taken from electron–positron collision data. Fermilab’s first result in 2021, combined with Brookhaven’s, lay 4.2 standard deviations above it. The same year the BMW collaboration computed that part with lattice QCD and found it closer to the measurement.

A new measurement by CMD-3 then left the collision data sets disagreeing with each other. The 2025 update adopted the lattice average instead; the prediction became 116 592 033(62) × 10⁻¹¹, and the difference from experiment 38(63) × 10⁻¹¹, no tension at the present precision. The measurement is about four times more precise than that prediction; the disagreement between lattice and collision data is still to be resolved.

*Experiment − theory = 38(63) × 10⁻¹¹ (2025 theory update, world average)*

## 15 Taken Apart

The yoke, the poles, the three coils and their cryostats, the vacuum chambers, the quadrupole plates, the kickers, the inflector and the beamline, the calorimeters, the trackers and the trolley, lifted apart layer by layer.

Rest on one and it is lifted out on its own while the rest fade; left alone, every part is named.

## 16 Another Field

The lever is the field. The muons’ momentum follows it, so they still run on the 7.112 m orbit. The wiggle plot at the top right is simulated and fitted as you watch by this page’s toy model; the slider is the kickers’ angle, and with the wrong kick no muons are stored.

A stronger field makes a faster wiggle: ω\_a is proportional to B. The electric term in the readouts is zero only near 1.45 T.

**Try this**: Set the field to 1.2 T: does the wiggle’s period get longer or shorter? Is the momentum in the readouts still the magic one?

## What the model leaves out

The ring follows the published cross-section of the magnet and the sizes in the design report, at about 1 : 15; the vacuum chambers’ flanges, bellows and ports, the calorimeters’ housings, racks and cables, the cable trays inside the ring, the trackers’ frames and the beamline outside the ring are schematic, not from engineering drawings, and so are the collimators’ places. In the real ring the quadrupoles, the inflector and the kickers sit inside the vacuum chambers; only their plates are drawn. The muons are drawn far too large and slowed about ten million times; to be seen at all, they also decay far faster than real ones, which run about 430 laps on average. Positron tracks are drawn as circles, ignoring the chamber walls and the calorimeters’ shapes. The wiggle plot comes from a toy model: muons all start together with spins along their momentum, decays are drawn from the Michel spectrum (with its parity-violating asymmetry) and boosted to the lab, and only positrons above 1.7 GeV are counted; it has no detector acceptance, pile-up, beam oscillations or muon losses, so its A is about 0.34, a little below the experiment’s 0.37. In the laboratory the momentum changes with the field; that is a thought experiment, as the real ring runs only at 1.45 T and the magic momentum.

## Some common misreadings

Fermilab’s result did not find new physics: the 2025 Standard Model prediction agrees with it at the present precision; what remains to be understood is why the theory’s two ways of computing the strong-interaction part disagree. Nor did “the whole 700-tonne magnet” travel in one piece: what moved whole were the 17-tonne superconducting coils and cryostats; the yoke, some 680 tonnes, was taken apart and moved separately. A muon’s spin is not a little ball spinning; “the spin runs ahead of the momentum” means the direction of its magnetic moment turns relative to its direction of motion. Positrons come out along the spin more often, not always; for μ⁻ the electrons go more often against the spin. And g − 2 is twice a\_μ: g − 2 = 2a\_μ ≈ 0.002 33.

## Common questions

### How precise is the final muon g−2 result?

On 3 June 2025 Fermilab’s Muon g−2 experiment announced its final result: the muon’s magnetic anomaly is a\_μ = 0.001 165 920 705, with a relative uncertainty of 127 ppb, 127 parts per billion, better than the 140 ppb the experiment set out to reach.

See: [01 127 Parts per Billion](https://world.lapla.si/en/muong2/#precision)

### Why is the muon’s g-factor not exactly 2?

virtual particles keep appearing and vanishing around any particle, and they push g a little above 2.

See: [02 g Is Not 2](https://world.lapla.si/en/muong2/#anomaly)

### How long does a muon live in the storage ring?

A μ⁺ decays into a positron and two neutrinos, after 2.197 µs on average at rest; in the ring it moves so close to the speed of light that its time runs 29.3 times slower, 64.4 µs on average, about 430 laps.

See: [05 Where the Positrons Go](https://world.lapla.si/en/muong2/#decay)

### Does the muon g−2 still disagree with theory?

The 2025 update adopted the lattice average instead; the prediction became 116 592 033(62) × 10⁻¹¹, and the difference from experiment 38(63) × 10⁻¹¹, no tension at the present precision.

See: [14 The Theory Side](https://world.lapla.si/en/muong2/#theory)

## Sources

1. [Muon g−2 Collaboration (2025): Measurement of the Positive Muon Anomalous Magnetic Moment to 127 ppb, PRL 135, 101802 (a\_μ = 116 592 0705(148) × 10⁻¹²; world average 116 592 0715(145) × 10⁻¹²)](https://arxiv.org/abs/2506.03069)
2. [Muon g−2 Collaboration (2026): Final Report on the Measurement of the Positive Muon Anomalous Magnetic Moment at Fermilab to 127 ppb, arXiv:2606.17323 (98 ppb statistical, 78 ppb systematic; six runs 2018–2023; “fully compatible” with the 2025 prediction)](https://arxiv.org/abs/2606.17323)
3. [Fermilab press release (3 June 2025, as reposted by Brookhaven): Muon g-2 announces most precise measurement of the magnetic anomaly of the muon (beyond the 140 ppb design goal)](https://www.bnl.gov/newsroom/news.php?a=222457)
4. [B. Abi et al. (2021): Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm, PRL 126, 141801 (4.2σ combined)](https://arxiv.org/abs/2104.03281)
5. [T. Albahri et al. (2021): Measurement of the anomalous precession frequency of the muon in the Fermilab Muon g−2 experiment, PRD 103, 072002 (the wiggle function; the 1.7 GeV threshold; 64.4 µs; 4.365 µs)](https://arxiv.org/abs/2104.03247)
6. [T. Albahri et al. (2021): Magnetic field measurement and analysis for the Muon g−2 experiment at Fermilab, PRA 103, 042208 (17 probes, about 9,000 places, petroleum jelly, the water probe, 14 ppm, 61.79 MHz)](https://arxiv.org/abs/2104.03201)
7. [T. Albahri et al. (2021): Beam dynamics corrections to the Run-1 measurement…, PRAB 24, 044002 (quadrupoles 13°/26°, 43 %, 18.3 kV; about 5,000 muons a fill; 16 fills in 1.4 s)](https://arxiv.org/abs/2104.03240)
8. [D. P. Aguillard et al. (2023): Measurement of the Positive Muon Anomalous Magnetic Moment to 0.20 ppm, PRL 131, 161802](https://arxiv.org/abs/2308.06230)
9. [D. P. Aguillard et al. (2024): Detailed report on the measurement … to 0.20 ppm, arXiv:2402.15410 (the insulating blanket; yokes and electronics within ±0.5 °C)](https://arxiv.org/abs/2402.15410)
10. [G. W. Bennett et al. (2006): Final report of the muon E821 anomalous magnetic moment measurement at BNL, PRD 73, 072003 (0.54 ppm; the magnet’s cross-section; the magic momentum 3.094 GeV/c; the CERN experiments)](https://arxiv.org/abs/hep-ex/0602035)
11. [J. Grange et al. (2015): Muon (g−2) Technical Design Report, arXiv:1501.06858 (12 sectors, about 680 t; poles 56 cm, tolerances; coil radii; 5.0 K)](https://arxiv.org/abs/1501.06858)
12. [G. T. Danby et al. (2001): The Brookhaven muon storage ring magnet, NIM A 457, 151](https://doi.org/10.1016/S0168-9002(00)00704-X)
13. [A. P. Schreckenberger et al. (2021): The fast non-ferric kicker system for the Muon g−2 experiment at Fermilab, NIM A 1011, 165597 (three 1.27 m magnets; 10.8 mrad; 77 mm)](https://arxiv.org/abs/2104.07805)
14. [K. S. Khaw et al. (2019): Performance of the Muon g−2 calorimeter and readout systems measured with test beam data, NIM A 945, 162558 (9 × 6 lead fluoride crystals of 25 × 25 × 140 mm)](https://arxiv.org/abs/1905.04407)
15. [A. T. Fienberg et al. (2015): Studies of an array of PbF₂ Cherenkov crystals with large-area SiPM readout, NIM A 783](https://arxiv.org/abs/1412.5525)
16. [B. T. King et al. (2022): The straw tracking detector for the Fermilab Muon g−2 experiment, JINST 17, P02035 (180° and 270°; 8 modules; 4 × 32 straws)](https://arxiv.org/abs/2111.02076)
17. [J. P. Miller, E. de Rafael, B. L. Roberts (2007): Muon (g−2): experiment and theory, Rep. Prog. Phys. 70, 795 (the outer coil’s two 24-turn windings; CERN’s third experiment)](https://arxiv.org/abs/hep-ph/0703049)
18. [T. Aoyama et al. (2020): The anomalous magnetic moment of the muon in the Standard Model, Phys. Rep. 887, 1 (116 591 810(43) × 10⁻¹¹)](https://arxiv.org/abs/2006.04822)
19. [Sz. Borsanyi et al. (2021): Leading hadronic contribution to the muon magnetic moment from lattice QCD, Nature 593, 51](https://arxiv.org/abs/2002.12347)
20. [R. Aliberti et al. (2025): The anomalous magnetic moment of the muon in the Standard Model: an update, Phys. Rep. 1143 (116 592 033(62) × 10⁻¹¹; difference 38(63) × 10⁻¹¹, “no tension”)](https://arxiv.org/abs/2505.21476)
21. [J. Schwinger (1948): On quantum-electrodynamics and the magnetic moment of the electron, Phys. Rev. 73, 416 (α/2π)](https://doi.org/10.1103/PhysRev.73.416)
22. [P. A. M. Dirac (1928): The quantum theory of the electron, Proc. R. Soc. A 117, 610](https://doi.org/10.1098/rspa.1928.0023)
23. [Particle Data Group: muon (mean life 2.196 9811(22) µs)](https://pdglive.lbl.gov/Particle.action?node=S004&init=0)
24. [NIST: CODATA 2022 fundamental physical constants (the muon’s mass, e, c)](https://physics.nist.gov/cuu/Constants/)
25. [Fermilab: The Big Move (2013, web archive: set off 22 June, arrived 4:07 a.m. 26 July, 35 days, 3,200 miles)](https://web.archive.org/web/20151231235836/http://muon-g-2.fnal.gov/bigmove/)
26. [Brookhaven news (29 July 2013): Giant electromagnet arrives at Fermilab (17 t; 60 t with its frame; the Tombigbee and Tennessee route)](https://www.bnl.gov/newsroom/news.php?a=24155)
27. [CERN Archives: Archives of Muon g-2 experiment (begun 1959; the 14 m storage ring)](https://scientific-info.cern/archives/CERN_archive/guide/experimental_physics/SC/isamuon)

## Data

The ring follows the published cross-section of the storage ring magnet (Bennett et al., PRD 73 (2006) 072003, fig. 9; Danby et al., NIM A 457 (2001) 151) and the sizes in the Fermilab design report (arXiv:1501.06858); the vacuum chambers’ flanges, bellows and ports, the calorimeters’ housings, racks and cables, the cable trays, the trackers’ frames and the beamline outside the ring are schematic. It is not built from engineering drawings. The yoke’s blue follows photographs of the ring at Fermilab, muted. The wiggle plot in the laboratory and the positron tracks are computed by this page from a toy simulation of the decay (the Michel spectrum boosted to the lab); they are not the experiment’s data. Material presets, parts and the softbox studio of the site’s model kit are ported from AirsupHQ/machine-skill (MIT, © 2026 Air Supply Technologies Inc.).

## Licence CC BY-NC 4.0

The text, pictures and models on this page are licensed CC BY-NC 4.0: free to use and adapt for teaching and other non-commercial purposes, with credit to world.lapla.si. Third-party material listed above keeps its own licence.

https://creativecommons.org/licenses/by-nc/4.0/deed.en

## Cite this plate

Laplace’s Mini World, Plate F38, “Muon g−2”, https://world.lapla.si/en/muong2/, CC BY-NC 4.0, updated 2026-10-22.